VECTOR ALGEBRA CHAPTER 12
Chapter Outline
12.1 Introduction to Vectors
12.2 Algebra of Vectors
12.3 Vectors in Rectangular Cartesian Coordinate System
12.4 Scalar Product of Two Vectors
12.5 Cross Product of Two Vectors
12.6 Multiple Products of Vectors
Vectors represent one of the most important mathematical systems, which is used to handle certain types of problems in Geometry, Mechanics, and other branches of Applied Mathematics, Physics, and Engineering.
Scalar and vector quantities: Those quantities which have only magnitude, and which are not related to any fixed direction in space are called scalar quantities, or briefly scalars. Mass, volume, density, work, temperature, etc are examples of scalar quantities. Those quantities that have both magnitude and direction are called vectors. Displacement, velocity, acceleration, momentum, weight, and force are examples of vector quantities.
12.1 INTRODUCTION TO VECTORS
The study of motion encompasses various quantities, such as distance, speed, force, and energy, among others. These can be categorised into vectors, described by magnitude and direction, and scalars, described solely by magnitude. This unit emphasizes understanding vector fundamentals to
analyse motion and forces in two dimensions, effectively. By grasping concepts like displacement, velocity, acceleration, and momentum, students gain insights into the intricacies of physical phenomena. Through application and comprehension of vector principles, learners acquire a comprehensive understanding of motion dynamics and the interplay of forces within the physical realm.
12.1.1 Physical Quantities
The physical quantities are divided into two categories:
1. Scalar quantities
2. Vector quantities
Scalar quantities: like mass, length, volume, density, time, and temperature, have only magnitude and no specific direction. They are real numbers, making them essential in measurements where direction isn’t a factor.
Force, velocity, acceleration, displacement, and momentum are Vector quantities, possessing both magnitude and direction. They are vital for comprehensive analysis in physics and engineering.
Quantities having magnitude and direction but not obeying vector law of addition will not be treated as vector quantity, e.g., rotation of a rigid body about a finite angle.
1. Classify the following measures as scalar and vectors.
1. 10 kg
2. 10 metres northwest
3. 10 newtons
CHAPTER 12: Vector Algebra 2
4. 30 kilometres per hour
5. 50 metres per second towards north.
6. 19 10 coulomb
Sol. Vector quantity is a physical quantity which has both magnitude and direction. Scalar quantity is a physical quantity which has magnitude and no direction.
Here,
1. Mass-scalar,
2. Directed distance-vector,
3. Force is vector,
4. Speed-scalar,
5. Velocity-vector,
6. Electric charge-scalar.
Try yourself:
1. Classify the following measures as scalars and vectors
1. 15 kg
2. 10 metres southeast
3. 50 metres per second
4. 45°
1,3,4Ans: – scalars, 2 – vector
12.1.2 Representation and Types of Vectors
The directed line segment with the initial point P and the terminal point Q is denoted by the symbol .PQ
2. Support: The line of unlimited length, of which a directed line segment is a part, is called the support.
3. Sense: The sense of the directed line segment is from its initial point to its terminal point.
We generally denote the vector by bold letter or by a single letter with an arrow or by a letter with a bar over its head, i.e., the vector PQ is denoted by a or a .
A vector whose magnitude is zero is called zero or null vector and it is represented by O
. For this vector, initial and terminal points are coincident, and its direction is indeterminate.
AABBPP
are zero vectors. Geometrically, zero vector is a point.
If the magnitude of a vector is one, then that vector is called a unit vector. If 1, a = then the vector a is unit vector, and it is denoted as . a
The unit vector in the direction of a is a a a =
The unit vector in the opposite direction of a is . a a
The unit vector parallel to a is . a a ±
The two end points are not interchangeable. A directed line segment is called a vector, if it has three following characteristics.
1. Length: The length of PQ
is denoted by PQ
If two or more vectors have the same initial point then those vectors are called coinitial vectors, e.g., ,,,, ABACADAE
etc.
If two or more vectors have the same terminal point then those vectors are called coterminal vectors. ,,, BACADAEA
are said to be coterminal vectors.
Collinear vectors are also called parallel vectors.
Collinear vectors are called like vectors if their directions are the same.
P R Q
Here, , PQQR
vectors are like vectors.
Collinear vectors are called unlike vectors if their directions are opposite.
P R Q
Here, and QPQR are unlike vectors.
All collinear vectors are parallel vectors and vice versa. Two non-zero vectors a and b are collinear or parallel vectors, if ab =λ .
Three or more vectors are said to be coplanar if they lie in the same plane, or they are parallel to the same plane. Three vectors ,,and abc are coplanar if and only if one can be expressed as linear combination of other two vectors.
. axbyc =+
If the origin of a vector is not specified, it is called a free vector. For a vector of given magnitude and direction, if its initial point is fixed in space, the vector is called a localised vector
Let O be the origin and P be any point in the space. The position vector of P is OP
. If a and b are position vectors of two points A and , B then ABOBOAba =−=−
.
The vector, which has the same magnitude as the vector a but the direction opposite to
that of a , is called the negative of a , and if is written as a . If PQa =
, then . QPa =−
Two vectors are said to be equal vectors if they have (i) the same length, (ii) same or parallel supports, (iii) the same sense. A C B D a b
Two zero vectors are equal vectors.
Two vectors with different directed line of support are not equal vectors.
Two vectors with different magnitudes are not equal vectors.
2. What is the unit vector along ? AB
Sol. The unit vector along AB
is
Try yourself:
2. In the figure below,
Scale 1 unit
which of the vectors are (i) collinear, (ii) equal, (iii) co-initial?
are co initial vectors.
and d
and d are collinear, (ii) a , c are equal vectors. (iii) , bc
, c
Ans: (i) a
TEST YOURSELF
1. Equal vectors have same (1) direction only (2) length only (3) length and direction (4) none of these
2. Which point of free vector is not specified?
(1) Initial point
(2) Terminal point (3) Midpoint
(4) None of these
3. Collinear vectors having opposite direction are called (1) unlike vectors (2) like vectors (3) co-inital vector (4) co-terminal vectors
4. The magnitude of a unit vector is (1) 1 (2) 0 (3) 2 (4) 3
Answer Key (1) 3 (2) 1 (3) 1 (4) 1
12.2 ALGEBRA OF VECTORS
In this section, we can create rules for adding, subtracting, this is called the algebra of vectors. This helps solve problems involving direction and magnitude in geometry, physics, and other areas of math.
12.2.1 Addition of Vectors
Let and ab be any two vectors in the space. Let O be a fixed point in the space. Suppose, OAa =
and ABb =
, so that the terminal point of the first vector is the initial point of the second vector. In a triangle OAB , OAa = and ABb =
Then, OBOAABab =+=+
. This is called triangle law of vector addition.
The triangle law of vector addition states that, if two vectors are represented by two sides of a triangle, then their resultant vector is represented by the third side in the opposite direction of a , b
.
Let a and b be two adjacent sides of a parallelogram whose initial point is one vertex O of it, as shown in the figure below. The diagonal of parallelogram having initial point at O is the sum of those two vectors.
In a parallelogram OACB , let OAa =
and OBb =
. Thus OCOAACOAOBab =+=+=+
.
This method of vector addition is called parallelogram law of addition of vectors.
and BCb =
Let ABCDEF be a hexagon such that ABa =
, as shown in the figure below.
From the figure:
1. ACABBCab =+=+
2. 22 ADBCb ==
3. CDADACba =−=−
4. DEABa =−=−
5. EFBCb =−=−
6. FACDab =−=−
7. 2 AEADDEba =+=−
8. 22 FCABa ==
In the above hexagon ABCDEF , . ABBCCDDEEFAF ++++=
This is called polygon law of vector addition.
Properties of Addition of Vectors
1. Addition of vectors is commutative, i.e., . abba +=+
2. Addition of vectors is associative, i.e., ()() .abcabc ++=++
3. Additive identity exists, i.e., 00. aaa +=+=
4. Additive inverse exists, i.e., ()()() 0. aaaa +−=−+=
3. If a , b , c are three sides of a triangle taken in order then show that 0. abc++=
Sol. Consider a triangle with sides ,,.abc
abcBCCAAB BAAB BB O ++=++ =+ = =
Therefore, abcO ++=
Try yourself:
3. If POOQQOOR +=+
, then show that the points ,, PQR are collinear.
12.2.2 Subtraction of Vectors
If a and b are two vectors, then their subtraction ab is defined as () , abab −=+−
where b is negative of b If 123 aaiajak =++ and 123 , bbibjbk =++ then ()()() 112233 ababiabjabk −=−+−+−
Properties of Vector subtraction:
1. Subtraction of vectors is not commutative, i.e., abba −≠−
2. Addition of vectors is associative, i.e., ()() abcabc −−=−−
3. Since any one side of a triangle is less than the sum of the other two sides, so, for any two vectors a and b , we have
i. abab +≤+
ii. abab +≥−
iii. abab −≤+
iv. abab −≥−
v. ababab −≤+≤+
4. Five forces ,,,, ABACADAE
and AF
are the vertices of a regular hexagon . ABCDEF Then, prove that
Sol. Given: ,,,, ABACADAE
are the vertices of a regular hexagon ABCDEF and O is centre of the hexagon. Let P be any point. Then,
3. ()()() mnanmamna ==
4. () mabmamb +=+
5. If a and b are two vectors, then write the truth value of the following statements.
1. If ab =− then ab =
2. If ab = then ab =±
3. If ab = then . ab =
Therefore,
Try yourself:
4. If P is a point, ABCD is a quadrilateral, and , APPBPDPC ++=
then prove that ABCD is a parallelogram.
12.2.3 Multiplication of Vector by a Scalar
If a is a vector and m is a scalar (i.e., a real number), then ma is a vector
(i) whose magnitude is m times that of a
(ii) whose direction is the same as that of a , if m is positive and opposite to that of a , if m is negative.
Properties of Multiplication of Vector by a Scalar
The following are properties of multiplication of vectors by scalars, for vectors a , b , and scalars ,.mn
1. ()()() mamama
Sol. Given: ab =− . It means that two vectors and ab are equal in magnitude but opposite in direction.
So, the statement I is true. It means, if , ab =− then . ab =
But ab = does not imply that the vectors and ab are in the same direction or in the opposite direction.
Hence, the other two statements, 2 and 3, are false.
Try yourself:
5. If a is a vector, then 0 ma = . What is the conclusions about , am Ans: Either a is null vector or 0 m =
TEST YOURSELF
1. The negative of () PQQR + vector is (1) PQ
(3) PR
(2) PR
(4) QR
2. In a trapezium ABCD , BCAD =λ and xACBD =+ . If , xPAD = then P = (1) 1 – λ (2) λ – 1 (3) 2 λ – 1 (4) λ + 1
3. If ABCDEF is a regular hexagon and , ABACADEAFAAB ++++=
then
is (1) 1 (2) 2 (3) 3 (4) 4
4. ABCDEF is a regular hexagon whose centre is ‘ O ’. Then, ABACADAEFA ++++
is equal to
12.3 VECTORS IN RECTANGULAR CARTESIAN COORDINATE SYSTEM
If the rotation from OX to OY is in the anti-clockwise direction and OZ is directed upwards, the system is called right handed system. O X Y Z Y Z X
2 AO
(3) AO
(2) 3 AO
(4) 6 AO
5. Let ABCDEF be a regular hexagon. If ADxBC = and CFyBA =
, then (x + y)2 + 8 = _____.
6. If D and E are the midpoints of AB and AC of ∆ABC and , DEBC =λ then 3 λ = ______.
7. Let ABCD be a parallelogram, and A1 and B 1 are the midpoints of sides BC and CD , respectively. If 11 AAABAC +=λ
. then 2 λ is equal to _____.
8. If D,E,F are the midpoinits of the sides BC,CA,AB, respectively, of a triangle ABC, and 211 33 ADBECFAC λ ++=
, then λ is _____.
Answer Key
(1) 2 (2) 4 (3) 4 (4) 4 (5) 24 (6) 6 (7) 3 (8) 2
If the rotation from OX to OY is in the clockwise direction and OZ is directed downwards, the system is called left handed system. Z Z X X Y Y O
12.3.1 Position vector of a point in space
Let O be a fixed point, known as the origin, and let ,, OXOYOZ be three mutually perpendicular lines, taken as x -axis, y -axis and z -axis, respectively, in such a way that they form a right handed system.
Let P be any point in space with coordinates () ,, lmn . The position vector of P is OPlimjnk =++
is 222 OPlmn =++
; the magnitude of OP
.
Here, ,, lmn are said to be components of the vector OPlimjnk =++
Components of a Vector in Terms of Coordinates of End Points
Let () 111 ,, Axyz and () 222 ,, Bxyz be any two points in the space. Then,
()()() 212121 .ABxxiyyjzzk =−+−+−
The magnitude of this vector is the distance between the two points A and B.
()()() 222 212121 ABxxyyzz =−+−+−
If aibjck ++ is a vector on a line of support L, then the direction ratios of that line is ,, abc and vice versa.
If limjnk ++ is a unit vector on a line of support L, then the direction cosines of that line L are ,, lmn or ,,. lmn
Addition, subtraction, and multiplication of a vector by a scalar in terms of components: For any two vectors 123 ˆ ˆ ˆ aaiajak =++ and 123 ˆ ˆ ˆ bbibjbk =++ , we can define:
1. ()()() 112233 ˆ ˆ ˆ ababiabjabk +=+++++
2. ()()() 112233 ˆ ˆ ˆ ababiabjabk −=−+−+−
3. ()()() 123 ˆ ˆ ˆ mamaimajmak =++
4. If ab = , then 121212 ,, aabbcc === and vice versa.
6. Find the values of x and y so that the vectors of ˆ ˆ ˆ 2 axijzk =++ and ˆ ˆ ˆ 2 biyjk =++ are equal.
Sol. If two vectors are equal, then their corresponding components are equal. It means that 2,2,1.xyz===
Try yourself:
6. If ˆ ˆ ˆ 23 aijk =++ and ˆ ˆ ˆ 245 bijk =+− represent two adjacent sides of a parallelogram, then find unit vectors parallel to the diagonals of the parallelogram.
Ans: )( 1 ˆ ˆ ˆ 362 7 ijk +− and )( 1 ˆ ˆ ˆ 28 69 ijk +−
12.3.2 Linear Combination of Vectors
A vector r is said to be a linear combination of the vectors ,,,...abc if there exist scalars ,,,...xyz such that ... rxaybzc=+++
If three vectors ,, rab are coplanar then one vector can be expressed as linear combination of other two vectors as rxayb =+
Let ,, abc be the position vectors of three points which are collinear. Then, there exist three scalars ,,,pqr not all zeros, such that 0 paqbrc++=
and 0. pqr++=
If ,,, ABCD are coplanar points then among three vectors ,,, ABACAD one vector can be expressed as linear combination of the other two vectors.
Collinear vectors are coplanar, but coplanar vectors may not be collinear.
If and ab are any two non-zero vectors, then , ab and null vector are coplanar.
Let ,,, abcd
be the position vectors of four points which are coplanar. Then, there exist four scalars ,,,,pqrs not all zeros, such that 0 paqbrcsd+++=
and 0 pqrs+++=
Let ,, abc be any three non-coplanar vectors. Then, any vector r can be expressed as linear combination of three non-coplanar
vectors. It means that there exist scalars ,,,xyz such that rxaybzc =++ .
7. Show that the vectors 23,abc −+ 35,abc −+ and 234 abc−+− are coplanar, where ,, abc are non-coplanar vectors. Sol. We must find the real numbers , xy such that
−+=−++−+−
Equate the corresponding components
21,332,543xyxyxy −=−+=−−=
Hence, 11 , 33xy==− . Clearly, these two values satisfy all three equations. Hence, the given three vectors are coplanar. Try yourself:
7. Show that the points 23,23,342 abcabcabc +−−++−
and 66 abc −+ are coplanar.
12.3.3 Linearly Dependent and Independent Vectors
A system of vectors 123,,,...aaa is said to be linear dependent if there exist scalars 123,,,...xxx (not all zeros), such that 112233 ,...0 xaxaxa++=
. A system of vectors 123,,,...aaa
is said to be linear independent if there exist scalars 123,,,...xxx (all zeros) such that 112233 ,...0 xaxaxa++=
.
Properties of Linearly Dependent and Linearly Independent Vectors:
1. Two non-zero, non-collinear vectors are linearly independent vectors.
2. Any two collinear vectors are linearly dependent vectors.
3. Any three non-coplanar vectors are linearly independent.
4. Any three coplanar vectors are linearly dependent vectors.
5. In the set of vectors, if zero vector is there, then that set is linearly dependent set of vectors.
6. Any four vectors in three-dimensional space are linearly dependent.
7. The condition that three points ()() 111222 ,,,,, AxyzBxyz , and () 333 ,, Cxyz are to be collinear is ()ABkAC = or 111 222 333 0 xyz xyz xyz = .
8. The condition that three vectors 123 ˆ ˆ ˆ aaiajak =++ , 123 ˆ ˆ ˆ bbibjbk =++ , and 123 ˆ ˆ ˆ ccicjck =++ are to be coplanar is 123 123 123 0 aaa bbb ccc = . In this case, one vector can be expressed as linear combination of other two. These three vectors are linearly dependent vectors.
8. If ,, abc are non-coplanar vectors, then show that the vectors 2 abc +− , 232 abc −+ , and 43 abc ++ are linearly independent vectors.
Sol. Suppose, there exists three scalars ,, xyz not all zeros, such that ()()() ()()() 2232430 2423 230. xabcyabczabc axyzbxyzcxyz +−+−++++= +++−++−++=
Equating the components to zeros, 240,230,230. xyzxyzxyz ++=−+=−++=
Solving the above three equations, we get 0,0,0xyz=== as only unique solution. Hence, the given vectors are not linearly dependent. They are linearly independent vectors.
Try yourself:
8. If ,,and abc are non-coplanar vectors, then show that the vectors 2 abc −+ , 2abc +− and 74abc −+ , are linearly dependent vectors.
12.3.4 Section Formula
If a point ()Rr divides the line segment joining the points ()Aa and ()Bb internally in the ratio : mn , then mbna r mn + = +
External Division
If a point ()Rr divides the line segment joining the points ()Aa and ()Bb externally in the ratio : mn , then mbna r mn = .
Midpoint Formula
If a point ()Rr divides the line segment joining the points ()Aa and ()Bb externally in the ratio 1:1 , then 2 ab r + = . If points ()Rr and ()Qq are trisecting points of the line segment joining the points ()Aa and ()Bb , and those points divide the line segment in the ratio 1:2 and 2:1, respectively, then, 2 3 ab r + = and 2 3 ab q + =
9. Find the midpoint of the line segment joining the points 32abc +− and 45.abc −+
Sol. Given: 32abc +− and 45 abc −+ are points The position vector of the midpoint of the line segment joining the above two points is 3245424 22 22 abcabcabcabc +−+−+−+ ==−+
Try yourself:
9. Find the ratio in which the point ˆ ˆ ˆ 23 ijk ++ divides the line segment joining the points
ˆ ˆ 235ijk−++ and ˆ ˆ 7. ik
12.3.5 Centres of Triangle
Ans: 1 : 2
In this section, we can find the formulae for the centroid, circumcentre, orthocentre, incentre of a triangle.
Centroid of a Triangle
Let ()() , AaBb and ()Cc are position vectors of vertices of a triangle ABC . Then, the position vector of centroid ()Gr is defined as () 3 abc Gr ++ =
If G is centroid of a triangle ABC , then 0 GAGBGC++=
and 3.OAOBOCOG ++=
If ,, DEF are midpoints of the sides ,, BCCAAB of the triangle , ABC then 0 ADBECF++=
and 3,ODOEOFOG ++=
where G is centroid of the triangle ABC .
Centroid of a Tetrahedron
Let ()()() ,, AaBbCc and ()Dd be position vectors of vertices of a tetrahed ron ABC
Then, the position vector of centroid ()Gr is defined as () 4 abcd Gr +++ =
Incentre of a Triangle
Let ()() , AaBb and ()Cc be position vectors of vertices of a triangle ABC Then, the position vector of incenter ()Ir is defined as () aabbcc Ir abc ++ = ++ where ,,.
Let OAa = and OBb = be any two vectors that are not parallel. The vector along the external angular bisector between a and b is kab ab
BCaCAbABc ===
Circumcentre of a Triangle
Let ()() , AaBb and ()Cc be position vectors of vertices of a triangle . ABC Then, the position vector of circumcentre ()Sr is defined as ()()()() sin2sin2sin2 sin2sin2sin2 AaBbCc Sr ABC ++ = ++
Orthocentre of a Triangle
Let ()() , AaBb and ()Cc be position vectors of vertices of a triangle ABC Then, the position vector of orthocentre ()Or is defined as ()()()() tantantan tantantan AaBbCc Or ABC ++ = ++
In triangle ABC, if S is circumcentre and O is orthocentre, then
i. SASBSCSO ++=
ii. 2 OAOBOCOS ++=
iii. 2 AOOBOCAS ++=
Angular Bisectors of Two Vectors
Let OAa = and OBb = be any two vectors that are not parallel. The vector along the internal angular bisector between a and b is kab ab
+
10. If ˆ ˆ ˆ 744 aijk =−− and ˆ ˆ ˆ 22 bijk =−−+ , then find the vector along c , which is along the internal angular bisector of a and b such that 56 c = units.
Sol. Given: ˆ ˆ ˆ 744 aijk =−− and ˆ ˆ ˆ 22 bijk =−−+
Angular bisector of a and b is ()()
491616414 11ˆˆ ˆˆˆˆ 74422 93
ˆ ˆ 72 9 ijkijk ijkijk ijk −−−−+ =+ ++++ =−−+−−+ −+ = The vector () 5 ˆ ˆ ˆ 72 3 cijk =±−+
Try yourself:
10. The vertices of a triangle are A (1,–1,–3), B (2,1,–2) and C (–5,2,–6). Then, find the length of the internal angular bisector of A
Ans: 310 4 units.
12.3.6 Straight Line and Plane
Equation of a line passing through a point ()Aa and parallel to the vector b is , ratb =+ where t is parameter. Equation of a line passing through two points ()Aa and ()Bb is ()ratba =+− where t is parameter.
12: Vector Algebra 12
The vector equation of a plane passing through a point ()Aa and parallel to the vectors b and c is ratbsc =++ . The vector equation of a plane passing through a points
()Aa , ()Bb and parallel to the vector c is () 1 rtatbsc =−++ . The vector equation of a plane passing through three points () , Aa
() , Bb and ()Cc is () 1 rstatbsc =−−++
To check if four points ()()()() ,,, AaBbCcDd are coplanar:
Step I: Find ,,. ABACAD
Step II: Find the determinant of the components. If it is zero, the points are coplanar; if not, they are not coplanar.
Position of a Point with Respect to the Triangle
Let O be the origin and , OAaOBb ==
be two vectors. The point OCpaqb =+
lies
■ inside the triangle OAB but inside the AOB∠ if 0,0pq>> and 1 pq+<
■ outside the triangle OAB but inside the AOB∠ if 0,0pq>> and 1 pq+>
■ outside the triangle OAB but inside the OAB∠ if 0,0pq<> and 1 pq+<
■ outside the triangle OAB but inside the OBA∠ if 0,0pq>< and 1 pq+<
11. Find the vector equation of the line passing through the points
Try yourself:
11. Find the vector equation of the plane passing through the point () 1,2,5 and parallel to the vectors () 6,5,1 and ()3,5,0.
ˆˆˆˆˆˆ 256535 rijksijktij =−++−−+−+ where , stR ∈
Ans: )()()(
TEST
YOURSELF
1. If AB = –3i + 4k and BC = –i – 2k represent two sides of ∆ ABC, then the length of the median through the vertex A is
(1) 45 2 (2) 45 2
(3) 85 2 (4) 85 2
2. If a = i+j – 2k, b = –i + 2j + k, c = i – 2j + 2k, then a unit vector parallel to a+b+c = (1) 2 6 ijk ++ (2) 3 ijk ++ (3) 2 6 ijk −+ (4) 3 ijk −+
3. A(4, 7, 8), B(2, 3, 4) and C(2, 5, 7) are the vertices of a triangle. Then, the position vector of the point of intersection of the internal bisector of ∠ A with the side BC is (1) () 2 68 6 3 ijk−++
(2) () 2 68 6 3 ijk ++
(3) () 1 613 18 3 ijk ++ (4) () 2 ijk ++
ˆ ˆ ijk ++ and ˆ ˆ
ijk −+
Sol. () ()() ˆˆˆˆˆˆ 1, rtijktijk =−+++−+ where t is any real number.
4. Find the position vectors of the points which divide the line joining the points 23ab and 32ab internally and e xt ernally in the ratio 2 : 3. (1) a (2) 5b (3) b (4) 5b
5. The unit vector(s) parallel to 35 ijk is
(1) () 1 35 35 ijk+−−
(2) 35 ijk
(3) () 1 35 35 ijk
(4) both 1 and 3
6. A vector a has components 3p and 1 with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, a has components p + 1 and 10 , then the value of p is equal to
(1) 5 4 (2) 4 5 (3) –1 (4) 1
7. Let , , abc be three non-coplanar vectors such that 123 , , , rabcrbcarabc =−+=+−=++
4 234. rabc =−+ If 4112233 , rprprpr =++ then p1 + p2 + p3 = _____.
8. If the vectors 1343, abcabc ++−++ and 2 abc µ ++ are linearly dependent, then μ = _____.
9. Given:
()()() 2015151212200 abpbcqcar −+−+−= when p,q,r are linearly independent, nonzero vectors and a,b,c are least possible integers. Then, value of c2 is ______.
Answer Key (1) 3 (2) 2 (3) 3 (4) 2 (5) 4 (6) 3 (7) 4 (8) 2015 (9) 25
12.4 SCALAR PRODUCT OF TWO VECTORS
If a and b are any two non-zero vectors represented by OA and ,OB respectively, then the angle between a and b is the angle between their directions when these directions both converge, or both diverge, from their point of intersection.
If θ is angle between two vectors, then the range of θ is 0,. π
If 2 θ=π , the vectors are said to be orthogonal or perpendicular to each other.
If 0, θ=π then the vectors are said to be parallel.
If the angle between a and b is θ , then () , ab =θ , () , ba =θ , (),.ab −−=θ
If () , ab =θ then () , ab−=π−θ and (),.ab −=π−θ
12.4.1 Definition of Dot Product of Two Vectors
If a and b are any two non-zero vectors, and θ is angle between them then the scalar or dot product of two vectors is denoted by ab ⋅ and is defined as cos abab⋅=θ .
Important points to remember
1. If either a or b is a null vector, then 0. ab⋅=
2. If 0 ab⋅= implies that either 0 a = or 0 b = or angle between a and b is 90. °
3. If θ is angle between two non-zero vectors a and b then cos. ab ab ⋅ θ=
4. If and ab are two non-zero vectors, and 0 ab⋅= , then a is perpendicular to . b
5. If the angle between two vectors is acute, then their dot product is positive.
6. If the angle between two vectors is obtuse, then their dot product is negative.
7. If a is a non-zero vector then 2 . aaa ⋅=
8. If abab ⋅= then the vectors a and b are parallel and like vectors.
9. If abab ⋅=− then the vectors a and b are parallel and unlike vectors.
10. For mutually perpendicular unit vectors, ˆ ˆ
,, ijk :
ˆˆ 1 iijjkk ⋅=⋅=⋅= and ˆˆ ˆˆ ˆˆ 0. ijjkki ⋅=⋅=⋅=
11. If 123 ˆ ˆ ˆ aaiajak =++
and 123
bbibjbk =++
then 112233 . abababab ⋅=++
12. Find ()() 32 abab +⋅−
when
ˆ ˆ 2 aijk =++ and
32. bijk =+−
Sol. Given:
2 aijk =++
and
32. bijk =+−
Try yourself:
12. Find the angle between the vec tors ˆ ˆ ˆ 534ijk ++ and ˆ ˆ ˆ 68. ijk +−
Ans: 1 2 cos 5101
12.4.2 Geometric Interpretation
Let and ab are two non-zero vectors and , OAaOBb ==
be two vectors. Suppose, the angle between a and b is (),.ab =θ O B A a P b q
The line segment OP is called the projection of ()OBb on the line ()OAa
Hence, the projection of b on a is cos ab OPOB a ⋅ =θ= ; this is also called component of b along . a
The projection of a on b is ab b ⋅ ; this is also called component of a along b . The dot product of two vectors is the product of magnitude of one of the vectors and the projection of the other vector on this vector.
Vector Components
ˆˆ 321075 105 15 ababijkik +⋅−=+−⋅−+
()()()()
1. Vector component of b in the direction of a is () 2 aba a
2. Vector component of b in the direction perpendicular to a is () 2 aba b a
13. Let ˆˆˆˆˆˆ ,23 aijkbijk =++=++ . Then, find projection vector of b on a and find its magnitude.
Sol. Given: Vectors are
,23. aijkbijk =++=++
The length of the projection of b on a is 231 23 111 ab a ⋅++ == ++
The vector component of b on a is 23 a a
Therefore, the required vector component is ˆ ˆ ˆ 222. ijk ++
Try yourself:
13. If P(1, 0 –1), Q(–1, 2, 0), R(2, 0, –3), S(3, –2, 1) then find the projection of RS on PQ
Ans: 4 3
12.4.3 Properties of Dot Product of Two Vectors
If a and b are two non-zero vectors and , pq are any two scalars, then
1. abab ⋅≤
2. ()()() ababab
3. ()()abab −⋅−=⋅
4. ()()() paqbpqab
Dot product of vectors obeys the distributive law from the right and from the left with respect to vector addition.
Let and ab and c be any three non-zero vectors. Then, ()bcabaca +⋅=⋅+⋅ and () abcabac ⋅+=⋅+⋅
Let and ab and c be any three non-zero vectors, and acbc ⋅=⋅ . Then, either ab = or () ab is perpendicular to c
If and ab and c are any three vectors, then
1. () 22 2 2 ababab +=++⋅
2. abab +≤+
3. () 22 2 2 ababab −=+−⋅
4. abab −≥−
5. ()() 2 2 ababab +⋅−=−
6. If () , ab =θ , then 2cos 2 ab θ +=
, where and ab are unit vectors.
7. If () , ab =θ , then 2sin 2 ab θ −=
, where and ab are unit vectors.
8. ababab +=−⇔⊥
9. 2222 222 abcabcabbcca ++=+++⋅+⋅+⋅
10. () 2 22 1 2 abbccaabc ⋅+⋅+⋅≥−++
11. 3 2 abbcca⋅+⋅+⋅≥− , where ,, abc are unit vectors.
12. If rxiyjzk =++ , then ()() () ... riirjjrkkr ++=
13. Angle between two diagonals of a cube is cos–1 (1/3).
14. Angle between diagonals of a cube and edge of a cube is 1 1 cos 3
15. Angle between diagonal of a cube and diagonal of a face of a cube is 1 2 cos 3
16. Cauchy-Schwartz inequality: I f a1, a2, a3, b1, b2, b3 are any real numbers, then (a1b1 + a2b2 + a3b3)2 ≤(a1 2 + a2 2 + a32) (b1 2 + b2 2 + b32)
14. Let ,, abc be three vectors such that 3,4,5abc=== , and () ab + is perpendicular to c , () bc + is perpendicular to a , and () ab + is perpendicular to c . Then, find the value of abc ++
Sol. Given: () 0 0, abcacbc +⋅=⇒⋅+⋅= () 00bcaabca +⋅=⇒⋅+⋅= and () 00cabcbab +⋅=⇒⋅+⋅=
Hence, 0 abbcca⋅+⋅+⋅=
2222 91625 50 abcabc ++=++ =++ =
Therefore, 52 abc++=
Try yourself:
14. Let ˆ ˆ ˆ 2 aijk =−+ , ˆ ˆ ˆ 2 bijk =+− , and ˆ ˆ ˆ 2 cijk =+− be the three vectors. If a vector of the type bc +λ has projection of magnitude 2 3 on a then find the value λ. 1Ans: or –3
12.4.4 Vector Equation of a Plane
The equation of plane at a distance p units from the origin and perpendicular to the unit vector ˆ n is ˆ rnp ⋅= .
The vector equation of plane passing through the point A with position vector a and perpendicular to the vector m is () 0. ram−⋅=
The perpendicular distance from origin to the plane () 0 ram−⋅= is . am m
If θ is the angle between two planes 1 rmp ⋅= and 2 rmq ⋅= then 12 12 cos mm mm ⋅ θ=
.
15. Find the perpendicular distance from origin to the plane ().2 3 6 21 rijk++=
Sol. 2121 Distance 3 7 4936 === ++
Try yourself:
15. Find the vector equation of the plane through the origin and perpendicular to the vector 235. ijk
Ans: )( . 2 3 5 0 rijk−−=
12.4.5 Physical Significance of Dot Product of Two Vectors
If a constant force F acting on a particle displaces it from a position A to the position B , then the work done W by this constant force F is the dot product of the vector representing the force F and displacement vector AB
. i.e., WFAB =⋅
If forces 123,,,... n FFFF
are constant forces acting on a particle, which is displaced from A to B , then the work done is () 123 n FFFFAB ++++⋅
.
If several forces are acting on a particle, the sum of the work done by the separate forces is equal to the work done by the resultant force.
16. A particle acted by constant forces ˆ ˆ ˆ 43ijk +− and ˆ ˆ ˆ 3ijk +− is displaced from a point () 1,2,3 to the point () 5,4,1 . Find the total work done by the forces in units.
Sol. Here, 12 ˆ ˆ ˆ 724. FFFijk =+=+−
Displacement vector is ˆ ˆ ˆ 422. ABijk =+− Hence, the work done is ()()
724422 2848 40 FABijkijk ⋅=+−+− =++ =
Therefore, the work done is 40.
Try yourself:
16. A particle acted by constant forces
ˆ ˆ ˆ 256ijk −+ and ˆ ˆ ˆ 2 ijk−+− is displaced from a point () 4,3,2 to the point () 6,1,3 . Find the total work done by the forces in units.
Ans: –15
TEST YOURSELF
1. If ,, abab + are unit vectors, then () , ab =
2. The angle between a diagonal of a cube and the diagonal of a face of the cube is (1)
3. If ,,and abc are unit vectors satisfying 2 22 ||9,abbcca −+−+−=
then
277 abc ++ is (1) 2 (2) 3 (3) 4 (4) 5
4. Let , AB , and C be vectors of length 3, 4, 5, respectively. Let ()()() ,,. ABCBCACAB ⊥+⊥+⊥+
Then, the length of () ABC ++ is (1) 22 (2) 32 (3) 42 (4) 52
5. If 23,2,3, aijkbijkcij =++=−++=+ and atb + is perpendicular to c , then | t | = _____.
6. If ,,ab and c are unit vectors satisfying 30abc−+= and the angle between a and c is K π . Then the value of 3K2 – 3K + 2 = ______.
7. If ,,ab and c are unit vectors satisfying 22 2 9 abbcca −+−+−= , then 255 abc ++ is_____.
8. Let G be centroid of the triangle ABC whose sides are of lengths a,b,c. If P is point in the plane of triangle ABC such that PA = 1, PB = 3, PC = 2, PG = 2, then (a2 + b2 + c2) = ______.
Answer Key
(1) 3 (2) 3 (3) 4 (4) 4 (5) 5 (6) 20 (7) 3 (8) 6
12.5 CROSS PRODUCT OF TWO VECTORS
In the previous section, we learnt dot product which is a scalar. In this topic, we shall define cross product of two vectors, which is a vector. The study of cross product is useful in finding a vector perpendicular to a plane, finding perpendicular distances to a line from a plane, as well as finding areas of certain geometrical plane figures.
12.5.1 Definition of Cross Product of Two Vectors
Let a and b be two non-zero, non parallel vectors, and () , ab =θ . The cross product of two vectors a and b is denoted by ab × and is defined as
sin. ababn ×=θ
Here, ˆ n
is a unit vector perpendicular to the plane determined by a and . b
17. Let a and b be any two vectors such that 2 3, 3 ab== . If ab × is unit vector, then find the angle between a and b
Sol. Given: ab × is unit vector. sin 2 13sin 3 1 sin 2 abab×=θ
Therefore, the angle between the vectors a and b is 45. °
Try yourself:
17. Let ,, abc be unit vectors such that 0 abac⋅=⋅= and the angle between b and c is 6 π . If ()anbc =× then find the value of .n Ans: ±2
12.5.2 Geometrical Meaning of Cross Product of Two Vectors
For any two non-zero vectors a and b , the vector ab × is a vector which is perpendicular to the plane determined by a and . b
The vector ab × gives the vector area of parallelogram whose adjacent sides are a and b .
Vector Areas
If either a or b is a null vector or a is parallel to b , we define 0. ab×= For any non-zero vector a , 0. aa×=
1. The vector area of the triangle ABC is () 1 2 ABAC ×
or () 1 2 BCBA ×
1 2 CACB ×
.
or
2. If ,,ab and c are the vertices of a triangle ABC , then the vector area of the triangle ABC is () 1 2 abbcca ×+×+×
.
3. The length of the altitude through A in triangle ABC is ABAC BC ×
4. Let a and b be diag onals of a parallelogram. Then, the vector area of the parallelogram is () 1 . 2 ab ×
18. Find the vector area of the triangle formed by the points ,,,ABC whose position vectors are
,23, OAijkOBijk =+−=−+
and
ˆ
32 OCijk =+− .
Sol. Given:
,23, OAijkOBijk =+−=−+
and
ˆ 32 OCijk =+−
The sides are
42 ABOBOAijk =−=−+
and
ˆ 2.ACOCOAik =−=−
Hence, the area of the triangle ABC is 1 2 ABAC ×
It implies that
Therefore, the area of the given triangle is 1
units. 22 ++=
Try yourself:
18. Find the unit vector perpendicular to the plane determined by the vectors ˆˆˆˆˆˆ 43,263. ijkijk +−−− Ans:
12.5.3 Properties of Cross Product of Two Vectors
The vector product of two vectors does not obey the commutative law. It means that for any two non-zero vectors, ()() abba ×=−×
But abba ×=×
If a and b
are any two non-zero vectors and and lm are scalars (real numbers), then
1. ()()ababab −×=×−=−×
2. ()()abab −×−=×
3. ()() lamblmab ×=×
Vector product, with respect to vector addition, obeys distributive law. It means that, for 3 vectors, () abcabac ×+=×+×
and () . abcacbc +×=×+×
Vector product among
1.
2.
ˆ ˆ ,,:ijk
ˆˆˆˆˆˆ ,, ijkjkikij ×=×=×=
ˆˆˆ ˆ ˆˆ ,, jikkjiikj ×=−×=−×=−
If 123 ˆ ˆ ˆ aaiajak =++
then ab ×
and 123 ˆ ˆ
bbibjbk =++
ˆ ijk abaaa bbb ×=
is defined as 123 123
This vector is perpendicular to both a and b
The unit vector perpendicular to both a and b is . ab ab × ×
If a is an vector then ()()() 222 2 2.aiajaka ×+×+×=
Lagrange’s identity: If a and b are any two non-zero vectors, then ()2 22 2 . ababab ×+⋅=
Moment of a force(Torque or Vector moment):
Let O be the point of referene and OPr = be the position vector of a point P on the line of action of a force F. Then, the moment of the fore F about O is given by . rF ×
19. If ()2 2 36 abab×+⋅= and 3 a = , then find the magnitude of b Sol. Using the Lagrange’s identity,
2 22 2 ababab ×+⋅=
Hence, 2 36 4 9 b ==
Therefore, 2 b =
Try yourself:
19. Find the unit vector perpendicular to both ˆ ˆ ˆ 23 aijk =++ and ˆ ˆ ˆ 22. bijk =+− Ans: 78 3 122 ij k
TEST YOURSELF
1. If 35, 63 aijbij =−=+ are two vectors and c is a vector, such that , caXb = , then :: abc =
(1) 34:45:39 (2) 34:45:39 (3) 34 : 39 : 45 (4) 39 : 35 : 34
2. If a is a non zero vector and .., , abacabac =×=× , then (1) ab (2) bc = (3) bc ≠ (4) and abac ≠=
3. The unit vector orthogonal to the plane determined by 22 ,3412 aijkbjjk =++=+− and forming a right-handed system with a and b is
(1) 28272 1517 ijk−++ (2) 28272ijk−++
(3) ijk ++ (4) jki +−
4. If 23 , 2 aijkbijk =+−=−+ , then a vector of length 5 and perpendicular to both and ab is (1) () 5 3 ijk±++ (2) () ijk±++
(3) () ijk±−+ (4) () 3 5 ijk±++
5. If , 23, aijkbijk =++=−+ then abba abba ×× += ××
(1) 0 (2) 22ijk +−
(3) 2 ijk ++ (4) 2 ijk +−
6. If and ABbACc == , then the length of the perpendicular from A to the line BC is (1) bc bc × + (2) bc bc ×
(3) ca ca × (4) ab ab ×
7. If 4590 abc++= , then ()()() abbcca××××× is equal to k Then k = ______.
8. If () , , then 45 ababba π =×+×= ____.
9. If () 2 2 2, 4, then 1cos, ab ab ab × === ____.
Answer Key
64
12.6 MULTIPLE PRODUCTS OF VECTORS
Earlier, we dealt with scalar product and vector product of two vectors. In this chapter, we extend the idea to product of three vectors and then to more vectors. Only some types of products of three vectors are meaningful. Certain products, like ()() , abcabc ⋅⋅⋅×
, have no meaning. Products like ()() , abcabc ×⋅××
, etc. Are meaningful.
Also, these products help us in providing conditions for the coplanarity of three vectors or four points. This part of the algebra of vectors is useful as tools while finding solutions of vector equations.
12.6.1 Scalar Triple Product of Vectors
Let ,, abc be any three vectors. Then, the quantity ()abc ×⋅ , which is a real number, is called scalar triple product of the vectors ,, abc . This can be represented by , abc read as box ,, abc .
1. The value of abc is zero if at least one of ,, abc is a null vector.
2. The value of abc is zero if ,, abc are coplanar vectors.
3. If ,, abc are three non-coplanar vectors, then 0 abc ≠ and vice versa.
4. Scalar triple product of three mutually perpendicular unit vectors is 1. It means, ˆ ˆ ˆ 1. ijk
Geometrical Meaning of Scalar Triple Product of Vectors
When ,, abc are non-coplanar vectors, then ()abc ×⋅ is interpreted geometrically as the volume of the parallelopiped with coterminous edges ,, abc . If ,, abc is a right-handed system of vectors, then abc
is volume of parallelopiped. If ,, abc is a left-handed system of vectors, then abc
is the volume of the parallelopiped.
The volume of the tetrahedron ABCD is 1 . 6 ABACAD
If position vectors of points O, A, B, and C of tetrehedron OABC are ,,, Oabc then position vector of circumcentre is ()()() 222 2 abcbcacab abc ×+×+×
Volume of a triangular prism, whose adjacent edges are along ,, abc , is 1 . 2 abc
Regular Tetrahedron
Let l be the edge of a regular tetrahedron.
i. Volume = 3 62 l
ii. Perpendicular distance from any vertex to the opposite face is 2 3 l .
iii. Angle between any two opposite edges is 90°.
iv. Angle between two adjacent faces is 1 1 cos 3
Condition
for
Coplanarity of Four Points
The condition for the points ,,, ABCD are to be coplanar is 0. ABACAD =
Properties of Scalar Triple Product of Vectors
1. If ,, abc are vectors, then ()()() abcbcacab ⋅×=×⋅=×⋅
2. If ,, abc are vectors, then ()() abcabc ⋅×=×⋅
3. If ,, abc are vectors, then paqbrcpqrabc =
4. If 123
aaiajak =++
and 123
ccicjck =++
, 123
bbibjbk =++
, then 123 123 123 aaa abcbbb ccc
5. If 0 a =
If two of ,, abc
, then 0.
are parallel, then 0. abc
6. If 123 aalaman =++
, 123 bblbmbn =++
and 123 ,cclcmcn =++ where ,, lmn
forms a right hand system of non-coplanar vectors, then 123 123 123 . aaa abcbbblmn ccc =
7. . lalblc lmnabcmambmc nanbnc
Vector Equation of Plane
The vector equation of plane passing through three points, whose position vectors are ,,,abc is rbcrcarababc
++=
The vector equation of the plane passing through a point whose position vector is () , Aa and which is parallel to the vectors b and , c is 0. rabc −=
The vector equation of plane passing through two points whose position vectors are ()() and AaBb
and parallel to the vector
c is 0 rabac −−=
The Shortest Distance between Two Skew Lines
The shortest distance between two skew lines
ratb =+
and rcsd =+ is . acbd bd ×
1. Distance between parallel lines 12 and rabrab =+λ=+µ is 12 (), . aab b −×
2. Let and ab be the vectors along the normals to the planes π 1 and π 2 , respectively. The vector ()aab ×× will be along the line of greatest slope in the plane π1.
The perpendicular distance from the origin to the plane containing three non-collinear
points ,, abc is abc abbcca ×+×+×
Let a and b be the lengths of two opposite edges of a tetrahedron, let d be their shortest distance, and let q be the angle between them. Then, the volume of tetrahedron is 1/6 abd sin q.
Angle between the Line and Plane
If θ is the angle between a line ratb =+ and the plane rnp ⋅= then sin
θ=
20. Find the value of λ for which the vectors ˆ ˆ ˆ aijk =−+ , ˆ ˆ ˆ 2 bijk =−+ , and
ˆ cijk =λ−+λ are coplanar.
Sol. If three vectors ,, abc are coplanar then 0. abc =
Hence, 111 2110 1 −= λ−λ
It implies that ()()() 1112120 120 1 −λ++λ−λ+−+λ= −λ++λ−+λ= λ=
Therefore, the given three vectors ,, abc are coplanar only when 1 λ=
Try yourself:
20. Find the volume of the tetrahedron having the coterminous edges ˆ ˆ ˆ 2ijk ++ , ˆ ˆ ˆ 32 ijk −+ and ˆ ˆ ik .
Ans: 2 cubic units
12.6.2 Vector Triple Product.
For the vectors ,, abc the products ()() , abcabc ×××× are called vector triple products.
The vector ()abc ×× is a null vector if any one of ,, abc is a null vector or , bc are parallel vectors, or a is perpendicular to the plane determined by ,.bc
A vector coplanar with , bc and perpendicular to a is () . abc ××
In general, ()() , abcabc ××××
are not equal vectors.
For any three vectors ,,.abc , ()()() abcacbabc ××=⋅−⋅
and ()()() . abcacbbca ××=⋅−⋅
21. Show that ()()() ˆˆ ˆˆ ˆˆ 2.iaijajkaka ××+××+××=
Sol. We know that ()()() abcacbabc ××=⋅−⋅
Hence, ()()()()
ˆˆˆˆˆˆˆˆ iaiiiaaiiaaii ××=⋅−⋅=−⋅ ()()()() ˆˆˆˆˆˆˆˆ jajjjaajjaajj ××=⋅−⋅=−⋅ and ()()()() ˆˆˆˆˆˆˆˆ .kakkkaakkaakk ××=⋅−⋅=−⋅
By adding all the above, we get ()()() ()()() ()()()
ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ 3 3 2 iaijajkak aaiiaajjaakk aaiiajjakk aa a ××+××+×× =−⋅+−⋅+−⋅ =−⋅−⋅−⋅ =− =
Therefore, ()()() ˆˆ ˆˆ ˆˆ 2 iaijajkaka ××+××+××=
12: Vector Algebra 24
Try yourself:
21. If ,, abc are three unit vectors such that () 1 2 abcb ××= then show that (),90ab =° and (),60.ac =°
12.6.3 Product of Four Vectors
If ,,, abcd are any four non-zero vectors, then ()() abcd ×⋅× is the scalar product of four vectors.
The scalar product of four vectors
()() abcd ×⋅×
can be expressed as
()()() ()() () . abcdacbdadbc ×⋅×=⋅⋅−⋅⋅
This can be remembered as . acad bcbd
If ,,, abcd
are any four non–zero vectors, then ()() abcd ××× is the vector product of four vectors.
()() abcdacdbbcda
22. If ()() 3 bccac ×××=
then find the value of bccaab
Sol. We know that
Try yourself:
22. Let ,, abc
are any three vectors,then show that (() )()() abccacbc ×××=⋅×
12.6.4 Reciprocal System of Vectors
If ,, abc
are three non-coplanar vectors, and if , bccaab abcabc ××′′ ==
and ab c abc ′× =
, then ,, abc′′′
are said to be reciprocal system of vectors for vectors ,, abc .
If the system of vectors ,, abc′′′
are orthogonal vectors of ,, abc then
1. 1 aabbcc ′′′ ⋅=⋅=⋅=
2. 0 bacbac ′′′ ⋅=⋅=⋅=
3. 1 abcabc ′′′ =
23. Let ,, abc be three non-coplanar vectors and ,, pqr be reciprocal system of vectors of ,,.abc
Then find the value of ()()() . abpbcqcar +⋅++⋅++⋅
Sol. Given: ,, pqr are reciprocal system of vectors to the system of vectors ,,.abc ()() 0 1 3 abpabbc abc abc abc ∑+⋅=∑+⋅×
Therefore, ()()() 3 abpbcqcar +⋅++⋅++⋅=
Try yourself:
23. If a1 , b1, c1 represents the reciprocal system of vectors ,, abc , then a a 1 , b b 1 , c c 1 = Ans: 3
TEST YOURSELF
1. Which of the following are meaningful?
(1) ()µνω × (2) ()µνω
(3) ()·· µνω (4) 1 or 3
2. If , , abc are non–coplanar vectors and ()()() 2 · , abcababcabc ++−×−−= λ then λ = (1) 3 (2) 5 (3) 7 (4) 8
3. For non-zero vectors , , , · abcabcabc ×= holds if and only if (1) ···0 abbcca===
(2) ···10 abbcca++=
(3) 0 abc++=
(4) 0 abc =
4. If ()() , abcdabdckd ×××=+ then k =
(1) bac (2) abc (3) bcd (4) cbd
5. If2 3, 2 , 4,aijkbijkcijk =+−=−+=−+− , dijk =++ then ()() abcd×××=
(1) 15 (2) 5114
(3) 114
(4) 5214
6. If a is a unit vector, then {()} aaab×××=
(1) ab × (2) ba × (3) () 2 ab × (4) () 2 ba ×
7. If ()() , abcabc ××=×× where , , abc are any three vectors such that .0, .0,abbc≠≠ then a and c are (1) parallel (2) inclined at an angle of 3 π between them (3) inclined at an angle of 6 π between them (4) perpendicular
8. ijkijkikk −+−−+−= ______.
9. If ()()() 222 1, , , 1, , , 1, , AaaBbbCcc === are non-coplanar vectors and 23 23 23 1 10 1 aaa bbb ccc + += + then | abc | = _____.
10. If , , abc are non-coplanar vectors and , 2, abcapbcabc ++++−++ are coplanar, then p = ______.
Answer Key
(1) 1 (2) 1 (3) 1 (4) 1 (5) 2 (6) 2 (7) 1 (8) 0 (9) 1 (10) 2
CHAPTER REVIEW
Introduction to Vectors
1. Scalar quantities have only magnitude and no specific direction.
2. Vector quantities possess both magnitude and direction.
3. In general, we denote the vector by bold letter or by a single letter with an arrow or by a letter with a bar over its head, i.e., the vector PQ
is denoted by a or a .
4. A vector whose magnitude is zero is called zero or null vector and its represented by O
, and its direction is indeterminate.
5. If the magnitude is a vector is one, then that vector is called a unit vector. If 1. a = then the vector a is unit vector, and it is denoted as a
6. The unit vector in the direction of a is
Algebra of Vectors
7. In a triangle OAB , OAa =
then OBOAABab =+=+
and ABb =
. This is called triangle law of vector addition
8. Let , ab are two adjacent sies of a parallelogram whose initial point is one vertex O of it as shown in the below figure. The diagonal of parallelogram having initial point at O is sum of those two vectors.
9. In a hexagon ABCDEF , ABBCCDDEEFAF ++++= , this is called polygon law of vector addition.
10. A vector r is said to be a linear combination of the vectors ,,,...abc if
there exist scalars ,,,...xyz such that rxaybzc=+++
11. If three vectors ,, rab are coplanar then one vector can be expressed as linear combination of other two vectors as rxayb =+
12. Let ,, abc be the position vectors of three points which are collinear, then there exists three scalars ,, pqr not all zeros, such that 0 paqbrc++= and 0. pqr++=
13. Let ,,, abcd be the position vectors of dour points which are coplanar, then there exists four scalars ,,, pqrs not all zeros, such that 0 paqbrcsd+++= and 0 pqrs+++= .
14. Let ,, abc
be any three non-coplanar vectors, then any vector r can be expressed as linear combination of three non-coplanar vectors... it means there exists scalars ,, xyz such that rxaybzc =++
15. A system of vectors 123,,,...aaa
is said to be linear dependent if there exists scalars 123,,,...xxx ( not all zeros) such that 112233 ,...0 xaxaxa++=
.A system of vectors 123,,,...aaa
is said to be linear independent if there exists scalars 123,,,...xxx (all zeros) such that 112233 ,...0 xaxaxa++=
.
16. Two non- zero, non collinear vectors are linearly independent vectors.
17. Any two collinear vectors are linearly dependent vectors.
18. Any three non-coplanar vectors are linearly independent.
19. Any three coplanar vectors are linearly dependent vectors.
20. In the set of vectors if zero vector is there then that set is linearly dependent set of vectors.
21. Any four vectors in three dimensional space are linearly dependent,
22. The condition that three points ()() 111222 ,,,,, AxyzBxyz and () 333 ,, Cxyz are to be collinear is
ABkAC = or 111 222 333 0 xyz xyz xyz =
23. The condition that three vec tors 123 ˆ ˆ ˆ aaiajak =++
, 123
bbibjbk =++
and 123 ˆ ˆ ˆ ccicjck =++ are to be coplanar is 123 123 123 0 aaa bbb ccc = , in this case one vector can be expressed as linear combination of other two. These three vectors are linearly dependent vectors.
24. Let ()() , AaBb and ()Cc are position vectors of vertices of a triangle ABC then the position vector of centroid ()Gr is defined as () 3 abc Gr ++ =
25. If G is centroid of a triangle ABC then 0 GAGBGC++=
and
3 OAOBOCOG ++=
26. If ,, DEF are midpoints of the sides ,, BCCAAB of the triangle ABC , then 0 ADBECF++=
and 3 ODOEOFOG ++=
where G is centroid of the triangle ABC .
27. Let ()()() ,, AaBbCc and ()Dd are position vectors of vertices of a tetrahedron ABC then the position vector of centroid ()Gr is defined as () . 4 abcd Gr +++ =
28. Let ()() , AaBb and ()Cc are position vectors of vertices of a triangle ABC then the position vector of Incenter ()Ir is defined as () aabbcc Ir abc ++ = ++ where ,, BCaCAbABc ===
29. Let ()() , AaBb and ()Cc are position vectors of vertices of a triangle . ABC Then the position vector of circum centre ()Sr is defined as ()()()() sin2sin2sin2 sin2sin2sin2 AaBbCc Sr ABC ++ = ++
30. Let ()() , AaBb and ()Cc are position vectors of vertices of a triangle . ABC Then the position vector of ortho entre ()Or is defined as ()()()() tantantan tantantan AaBbCc Or ABC ++ = ++
+
31. Let OAa = and OBb = be any two vectors which are not parallel, the vector along the internal angular bisector between a and b is kab ab
32. Let OAa = and OBb = a re any two vectors which are not parallel, the vector along the external angular bisector between a and b is . kab ab
33. Eq uation of a line passing through a point ()Aa and parallel to the vector b is ratb =+ where t is parameter.
34. Equation of a line passing through two points ()Aa and ()Bb is () . ratba =+− where t is parameter.
The vector equation of a plane passing through a point ()Aa and parallel to the vectors b and c is ratbsc =++
35. The vector equation of a plane passing through a points ()Aa , ()Bb and parallel to the vector c is () 1 rtatbsc =−++ .
36. The vector equation of a plane passing through three points ()Aa , ()Bb and ()Cc is () 1 rstatbsc =−−++
37. Let O be the origin , OAaOBb ==
be two vectors, the point OCpaqb =+ lies
38. Inside the triangle OAB but inside the AOB∠ if 0,0pq>> and 1 pq+<
39. Outside the triangle OAB but inside the AOB∠ if 0,0pq>> and 1 pq+>
40. Outside the triangle OAB but inside the OAB∠ if 0,0pq<> and 1 pq+<
41. Outside the triangle OAB but inside the OBA∠ if 0,0pq>< and 1 pq+<
Scalar Product of Two Vectors
42. If θ is angle between two vectors, then the range of θ is 0,. π
43. If the angle between a and b is θ , then () , ab =θ , () , ba =θ , (),.ab −−=θ
44. If () , ab =θ then () , ab−=π−θ and () , ab −=π−θ
45 If a and b are any two non-zero vectors, and θ is angle between them then the scalar or dot product of two vectors is denoted by ab and is defined as
cos abab⋅=θ .
46. If either a or b is a null vector, then 0 ab⋅= .
47. If 0 ab⋅= implies that either 0 a = or 0 b = or angle between a and b is 90° .
48. If θ is angle between two non zero vectors a and , b then cos ab ab θ=
49. If the angle between two vectors is acute then their dot product is positive.
50. If the angle between two vectors is obtuse then their dot product is negative.
51. If a is a non zero vector, then 2 aaa ⋅=
52. If , abab ⋅= then the vectors a and b are parallel and like vectors.
53. If abab ⋅=− then the vectors a and b are parallel and unlike vectors
54. For mutually perpendicular unit vectors, ˆ ˆ ˆ ,, ijk : ˆˆ ˆˆ ˆˆ 1 iijjkk ⋅=⋅=⋅= and ˆˆ ˆˆ ˆˆ 0. ijjkki ⋅=⋅=⋅=
55. If 123 ˆ ˆ ˆ aaiajak =++ and 123 ˆ ˆ ˆ , bbibjbk =++ then 112233 abababab ⋅=++
56. The projection of b on a is cos ab OPOB a ⋅ =θ= , this is also called component of b along . a
57. The projection of a on b is ab b , this is also called component of a along b .
58. Vector component of b in the direction of a is () 2 . aba a
59. Vector component of b in the direction perpendicular to a is () 2 . aba b a
60. Dot p roduct of vectors obeys the distributive law from the right and from the left with respect to vector addition.
61. The equation of plane at a distance p units from the origin and perpendicular to the unit vector ˆ n is ˆ rnp ⋅= .
62. The vector equation of plane passing through the point A with position vector a and perpendicular to the vector m is () 0 ram
63. The perpendicular distance from origin to the plane () 0 ram−⋅=
is am m
64. If a constant force F acting on a particle displaces it from a position A to the position B , then the work done W by this constant force F is the dot product of the vector representing the force F and displacement vector AB .It means, .WFAB =⋅
65. If forces 123,,,... n FFFF
are constant forces act on a particle which is displaced from A to , B then the work done is () 123 n FFFFAB ++++⋅
66. If several forces are acting on a particle, the sum of the work done by the separate forces is equal to the work done by the resultant force.
Cross Product of Two Vectors
67. Let a and b be two non-zero, no parallel vectors and () , ab =θ . The cross product of two vectors a and b is denoted by ab × and is defined as ˆ sin. ababn ×=θ
68. Here ˆ n is a unit vector perpendicular to the plane determined by a and . b
69. If either a or b is a null vector or a is parallel to b , we defined 0. ab×=
70. For any non-zero vector a , 0. aa×=
71. For any two non-zero vectors a and , b the vector ab × is a vector which is perpendicular to the plane determined by a and . b
72. The vector ab × gives the vector area of parallelogram whose adjacent sides are a and b .
73. The vector area of the triangle ABC is () 1 2 ABAC ×
or () 1 2 BCBA ×
or () 1 2 CACB ×
74. If , ab and c are vertices of a triangle ABC then the vector area of the triangle ABC is () 1 2 abbcca ×+×+×
75.The length of the altitude through A in triangle ABC is ABAC BC ×
76. Le t a and b are diagonals of a parallelogram, then the vector area of the parallelogram is () 1 . 2 ab ×
82. If ,, abc are three non-coplanar vectors, then 0 abc ≠ and vice versa.
83. Scalar triple product of three mutually perpendicular unit vectors in righthanded system is 1, it means
1. ijk =
84. The volume of the tetrahedron ABCD is 1 . 6 ABACAD
85. The condition for the points ,,, ABCD are to be coplanar is 0. ABACAD =
86. If ,, abc are vectors, then ()()() abcbcacab ⋅×=×⋅=×⋅
87. If ,, abc
77. If 123 ˆ
ˆ aaiajak =++
and 123
then ab × is defined as
this is a vector perpendicula r to both a and . b
78. The unit vector perpendicular to both a and b is ab ab × ×
79. Lagrange’s Identity: If a and b are any two non-zero vectors, then ()2 22 2 ababab ×+⋅=
Multiple Products of Vectors
80. The value of abc is zero, if at least one of ,, abc is a null vector.
81. The value of abc is zero, if ,, abc are coplanar vectors.
are vectors, then ()() abcabc ⋅×=×⋅
88. If ,, abc are vectors, then paqbrcpqrabc =
89. If 123 ˆ ˆ ˆ aaiajak =++
and 123
, 123 ˆ ˆ ˆ bbibjbk =++
ˆ ˆ ccicjck =++ then 123 123 123 . aaa abcbbb ccc =
90. If 123 aalaman =++
, 123 bblbmbn =++
and 123 cclcmcn =++
where ,, lmn forms a right hand system of noncoplanar vectors, then 123 123 123 aaa abcbbblmn ccc =
2 aaabac abcbabbbc cacbcc
91. The vector equation of plane passing through three points whose position vectors ,, abc is rbcrcarababc
92. The vector equation of plane passing through a point whose position vector is ()Aa and parallel to the vectors b and
c is 0 rabc
93. The vector equation of plane passing through two point whose position vectors are ()() , AaBb and parallel to the vector
c is 0 rabac
94. The shortest distance between two skew lines ratb =+ and rcsd =+ is acbd
95. If θ is the angle between a line ratb =+ and the plane ,rnp ⋅= then sin
θ=
96. For the vectors ,, abc the products ()() , abcabc ×××× are called vector triple products.
97. The vector ()abc ×× is a null vector if any one of ,, abc is a null vector or , bc are parallel vectors, or a is perpendicular to the plane determined by ,.bc
98. The vector ()abc ×× is a vector coplanar with , bc and perpendicular to the vector . a
99. In general, ()() , abcabc ×××× are not equal vectors.
100. For any three vectors ,, abc , ()()() abcacbabc ××=⋅−⋅ and ()()() abcacbbca ××=⋅−⋅
101. If ,,, abcd are any four non-zero vectors, then ()() abcd ×⋅× is the scalar product of four vectors.
102. The scalar product of four vectors ()() abcd ×⋅× can be expressed as ()()()()()() abcdacbdadbc ×⋅×=⋅⋅−⋅⋅ this can be remembered a s acad bcbd
103. If ,,, abcd are any four non-zero vectors, then ()() abcd ××× is the vector product of four vectors.
()() abcdacdbbcda ×××=−
or ()() . abcdabdcabcd ×××=−
104. If ,, abc
are three non-coplanar vectors, and let , bccaab abcabc ××
and ab c abc ′× =
, then ,, abc′′′
are said to be reciprocal system of vectors for vectors ,, abc
.
Exercises
JEE MAIN LEVEL
Level - I
Algebra of Vectors
Single Option Correct MCQs
1. If the vectors 236 aijk =++ and b are collinear and 21, b = then b equals to (1) () 236ijk±++ (2) () 6918ijk±++ (3) () 21 3 ijk ++
(4) ()21236ijk±++
2. Let α, β , γ be distinct real numbers. The points with position vectors
, ijkijk αβγβγα ++++ and
ˆ ijk γαβ ++ (1) are collinear (2) form an equilateral triangle (3) form a scalene triangle (4) form a right–angled triangle
3. The vector cosαcosβi + cosαsinβj + sinαk is a: (1) null vector (2) unit vector (3) constant vector (4) None of these
4. Let , and abc be three nonzero vectors, no two of which are collinear. If the vector 2 ab + is collinear with c , and 3 bc + is collinear with , then 2 6 aabc ++ = (1) aλ (2) bλ (3) cλ (4) O
5. The position vectors of three points A,B and C are (1, 3, x), (3, 5, 8) and (y, –1, –6) respectively. If A, B and C are collinear, then (x,y) =
(1) 2 , 3 3
(2) 10 , 3 3
(3) 10 , 3 3
(4) 10 3, 3
6. If the points (5, 4, 2), (8, K, –7) and (6, 2, –1) are collinear, then K = (1) –2 (2) 2 (3) 10 (4) 1
7. If ABCDEF be a regular hexagon in the xy plane and 4 ABi = then CD = (1) 623ij + (2) ()23ij−+ (3) ()23ij + (4) ()23ij
8. If 4 ˆ ˆ , ˆ ˆ pijqkj =+=− and ˆ ˆ rik =+ then the unit vector in the direction of 3 p + q – 2r is (1) () ˆ 1 ˆ 2 ˆ 2 3 ijk ++ (2) () ˆ 1 ˆ 2 ˆ 2 3 ijk (3) () ˆ 1 ˆ 2 ˆ 2 3 ijk −+ (4) ˆ 2 ˆ 2 ˆ ijk ++
9. If the points with position vectors 1013, ijkα++ 61111 ijk ++ , 9 8 2 ijk β +− are collinear then (19α – 6β)2 is equal to (1) 16 (2) 36 (3) 25 (4) 49
10. Let a and b be non–collinear vectors. If the vectors ()12 and 3 abab λλ −++ are collinear, then the set of all possible values of λ is (1) {2, 3} (2) {–2, 3} (3) {–2, –3} (4) {2, –3}
11. The vector that must be added to ˆ 2 ˆ 3 ˆ ijka −+ and ˆ 37 ˆ 6 ˆ ijk +− so resultant vector is a unit vector along the x-axis is
(1) ˆ 45 ˆ 2 ˆ ijk ++ (2) 5 ˆ ˆ ˆ 42ijk−++
(3) ˆ 35 ˆ 4 ˆ ijk ++ (4) Null vector
12. The unit vector parallel to the resultant vector of 4813,23 ijkijk +−++
(1) 1 22 3 ijk +−
(3) 1 22 3 ijk −++
is
(2) 1 22 3 ijk
(4) 1 23 3 ijk
Numerical Value Questions
13. If the points with position vectors 603,408,52 ijijaij +−−
are collinear then | a | = ______.
14. The points with position vectors 10 3,15 ˆˆˆˆ 2 ijij +− , and 11 ˆ ˆ aij + are collinear if a equals _____.
15. If 236,623,362 ijkijkijk +−−+−− represents the sides of a triangle then the perimeter of the triangle is _____.
Vectors in Rectangular Cartesian Coordinate System
Single Option Correct MCQs
16. Find the position vectors of the points which divide the line joining the points 23ab and 32ab internally and externally in the ratio 2 : 3.
(1) a (2) 5b
(3) b (4) 5b
(1) () 1 4517 3 ijk−++ (2) () 1 4517 3 ijk −+ (3) () 1 4517 3 ijk +− (4) None
19. If G(2, –1, 2) is the centroid of a tetrahedron OABC where O is (0, 0, 0) and G 1 is the centroid of ∆ ABC, then 1OG
is
(1) 1 (2) 3 2 (3) 4 (4) 9 2
20. A(4, 7, 8), B(2, 3, 4) and C(2, 5, 7) are the vertices of a triangle. Then the position vector of the point of intersection of the internal bisector of ∠ A with the side BC is (1) () 2 686 3 ijk−++
(3) () 1 61318 3 ijk ++
(2) () 2 686 3 ijk ++
(4) () 2 ijk ++
21. Let ABC be a triangle whose circumcentre is at P. If the position vectors of A,B,C and P are ,, abc and 4 abc ++ respectively, then the position vector of the ortho–centre of this triangle, is (1) abc ++ (2) O (3) 2 abc ++ (4) 3 abc ++
22. If O is any point such that OAOBOCODxOE +++=
17. G is the centroid of the triangle ABC and if G , is the centroid of another triangle A1B1C1, then value of 111 AABBCC ++
(1) 1GG (2) 0 (3) 1 3 GG (4) None of these
is :
18. The position vectors of A,B,C are respectively i + 2 j + 3 k , – i - j + 8 k , –4 i + 4 j + 6 k . Then the position vector of the incentre of ∆ ABC is
and ABCD is quadrilateral. If E is the point of intersection of the line joining the mid–points of opposite sides then x = (1) 1 (2) 2 (3) 3 (4) 4
23. If (1, 5, 35), (7, 5, 5), (1, λ, 7) and (2λ, 1, 2) are coplanar, then the sum of all possible values of λ is (1) 44 5 (2) 39 5 (3) 44 5 (4) 39 5
24. The lines
()()() 664448 rsasbsc =−+−+− and ()()() 214223 rtatbtc =−+−−+
intersect at
(1) 4 c (2) 4 c (3) 3 c (4) 2 c
25. If P is a point on the line parallel to the vector 236ijk and passing through the point A whose position vector 22 ijk +− and AP = 21, then the position vector of P can be
(1) 6918ijk (2) 6918ijk +−
(3) 51116 ijk−++ (4) 51116 ijk −+
26. If I is the centre of a circle inscribed in a triangle ABC, then
BCIACAIBABIC ++ is
(1) 0 (2) IAIBIC ++
(3) 3
IAIBIC ++ (4) 2
IAIBIC ++
27. The median AD of the triangle ABC is bisected at E.BE meets AC in F then AF : AC is
(1) 3 : 4 (2) 1 : 3
(3) 1 : 2 (4) 3 :2
28. If 745ijk −+ is the position vector of the vertex A of a tetrahedron ABCD and 43 ijk−+− is the position vector of the centroid of the triangle BCD , then the position vector of the centroid of the tetrahedron ABCD is
(1) 43 ijk−+− (2) 1 43 2 ijk−+−
(3) 2 ijk +− (4) 2 ijk−−+
29. Let the vectors 22 ABijk =++ and 244 ACijk =++ be two sides of a triangle ABC . If G is the centroid of ∆ ABC , th en () 2 27 5 7 AG +=
(1) 25 (2) 38 (3) 47 (4) 52
Numerical Value Questions
30. The number of distinct real values of λ , for which the vectors 22 , ijkijkλλ −++−+ and 2 ijk +−λ are coplanar is ______.
31. If a1 and a2 are two values of a for which the unit vector 1 ˆ 2 ˆ ˆ aibjk ++ is linearly dependent with ˆ 2 ˆ ij + and ˆ 2 ˆ jk , then 12 11 11 aa + is equal to _____.
32. Let a,b,c are three non-coplanar vectors such that 123,,, rabcrbcarcab =−+=+−=++
234, If , rabcrrrr λλλ=−+=++ and λ1 + λ2 + λ3 = k, then the value of k is ______.
112233
33. If elimjnk =++ is unit vector then maximum value of lm+mn+nl = ______.
34. The number of distinct real values of λ for which the vectors 33 , ˆˆˆˆ iKijλλ +− and () n ˆ ˆ ˆ 2si ijk λλλ+−− are coplanar is _____.
35. Let ABC be a triangle whose centroid is G, orthocentre is H and circumcentre is the origin ‘O’. If D is any point in the plane of the triangle such that no three of O,A,C and D are collinear satisfying the relation AD+BD+CH+3HG = λ HD. Then what is the value of the scalar λ ?
36. The position vectors of the vertices of a triangle are 345,7 ijkik +++ and 55ij + . The distance between the circumcentre and orthocenter is k . Then k is equal to ____.
Scalar Product of Two Vectors
Single Option Correct MCQs
37. Let a and b be unit vectors inclined at an angle 2 α , (0 ≤ α ≤ π) each other, then 1, ab+< if (1) 2 π α= (2) 3
(3)
38. Let ,, abc be pair wise mutually perpendicular vectors, such that 1,2,2.abc=== Then length of abc ++ is equal to (1) 3 (2) 4 (3) 2 (4) 6
39. ,and abc are unit vectors such that 34abc++= angle between a and b is θ1, angle between b and c is θ2 and angle between a and c varies 2 ,. 63 ππ Then the maximum value of cosθ1 + 3cosθ2 is (1) 3 (2) 4 (3) 2 (4) 6
40. The values of ‘’c’’, so that for all real x, the vectors 63,22 cxijkxijcxk −+++ makes an obtuse angle are (1) c < 0 (2) 4 0 3 c << (3) c > 0 (4) 4 0 3 c <<
41. A,B,C,D are four points with position vectors , , , abcd such that ()()()().0.adbcbdca −−=−−=
The point D is the______of ∆ ABC. (1) orthocentre (2) centroid (3) incentre (4) circumcentre
42. If ,and abc are vectors such that 1. and 23 bc abc === are perpendicular, and projections of and bc on a are equal then abc−+= (1) 4 (2) 14 (3) 23 (4) 10
43. ,and abc are three vectors of equal magnitude. The angle between eac h pair of vectors is 3 π such that 6 abc++= , then a is equal to (1) 2 (2) –1 (3) 1 (4) 6 3
44. In a ∆ ABC , , , OAaOBbOCc ===
If ()().0 , abcb−−= then P.V . of the circumcentre is (1) 2 ab + (2) 2 bc + (3) 2 ac + (4) 2 abc ++
45. If a right angled triangle ABC, the hypotenuos AB = p, then . . . ABACBCBACACB ++= (1) p (2) p2 (3) p3 (4) p4
46. The values of x for which the angle between the vectors 2 24 ˆ ˆ ˆ xixjk ++ and ˆ 7 ˆ ˆ 2 ijxk −+ are obtuse and the angle between the z–axis and ˆ 7 ˆ ˆ 2 ijxk −+ is acute and less than 6 π is given by :
(1) 1 0 2 x <<
(2) 1 2 x > or x < 0
(3) 1 15 2 x <<
(4) There is no such value for x
47. The vector equation of the plane which is perpendicular to 236ijk ++ and at a distance of 7 units from origin is
(1) ().2 3 6 72 rijk++=
(2) ().2 3 6 49 rijk++=
(3) ().2 3 614rijk++=
(4) ().2 3 6 49 rijk+−=
48. The angle between the planes
()() . 2 5, . 2 3 rijkrijk++=−+= is
(1) 30° (2) 45° (3) 60° (4) 90°
49. Equation of the plane passing through the point (3, 4, 5) and par allel to the plane
().2 3 6 rijk+−= is
(1) () . 2 3 13 rijk+−=−
(2) () . 2 3 13 rijk+−=
(3) () . 2 3 14 rijk+−=−
(4) () . 2 3 13 14 rijk +− =
50. The ⊥ bisecting plane of the line segment joining the points (3, 1, 2), (5, –3, 4) is
(1) ().2 4 2 4 rijk−+=
(2) () . 2 6 2 rijk−+=
(3) () . 2 9 rijk−+=
(4) () . 2 rijk++=
51. If forces of magnitudes 6 and 7 units acting in the directions 2 2 ijk −+ and 2 3 6 ijk respectively act on particle which is displaced from P(2, –1, –3) to Q(5, –1, 1) then the work done by the forces is (1) 4 units (2) –4 units (3) 7 units (4) –7 units
52. A particle is acted upon by constant forces 4 3 ijk +− and 3ijk +− which dis–place it f rom a point 2 3 ijk ++ to the point 5 4 .ijk ++ The work done in standard unity by the forces is given by (1) 40 (2) 15 (3) 25 (4) 30
53. The acute angle between ()() 3236 rikijk λ =−++++ and () ·102113 rijk+−= is (1) 1 8 sin 21
(3) 1 5 sin 21
(2) 1 8 cos 21
(4) 1 5 cos 21
54. The angle between any two opposite pair of edges of a regular tetrahedron (i.e all faces are equilateral triangles) is (1) 90° (2) 60° (3) 120° (4) 45°
Numerical Value Questions
55. Let 12 , ee be unit vectors which include an angle q . If () 12 1 sin, 2 eek−=θ then k = _____.
56. If 2 3 , 2 , 3 aijkbijkcij =++=−++=+ and atb + is perpendicular to c , then t = _____.
57. The perpendicular distance of a corner of unit cube from a diagonal not passing through it is k. Then 3k2 = _____.
58. Let 435ijk α ∧∧∧ =++ and 24 ijk β ∧∧∧ =+−
Let 1β be parallel to α and 2β is perp endicular to α . If 12 , βββ =+ then the value of 2 5. ijk β ∧∧∧
++=
____.
59. , ab and c are perpendicular to , and bccaab +++
, respectively, and if 6,8 and 10 abbcca +=+=+=
then abc ++
is equal to _____.
Cross Product of Two Vectors
Single Option Correct MCQs
60. If a + 2b + 3c = 0 and (a × b) + (b × c) + (c × a) = λ(b × c), then the value of λ is equal to (1) 3 (2) 4 (3) 6 (4) 2
61. Let 23,aijk =++ 2 bijk =−+ and 533 cijk =−+ be three vectors. If r is a vector such that, rbcb ×=× and .0ra = , then 2 25 r is equal to (1) 449 (2) 336 (3) 560 (4) 339
62. For any vector ()()()() 2 2 2 2 ˆ ˆ , ˆ aaiajaka ×+×+×=λ then λ = (1) 2 (2) 3 (3) 4 (4) 8
63. If , uabvab =−=+
and 2, ab==
then uv×=
(1) ()2 216 ab (2) ()2 16.ab (3) ()2 24.ab (4) ()2 4.ab
64. The point of intersection of , rabarbab ×=××=× where ,2 aijbik =+=− is (1) 3 ijk +− (2) 3 ik
(3) 32ijk ++ (4) None
65. For any four points P,Q,R,S, PQRSQRPSRPQS ×−×+× is equal to 4 times the area of the triangle (1) PQR (2) QRS (3) PRS (4) PQS
66. If x and y are two non–collinear vectors and ABC is a triangle with sides a,b,c satisfying ()()() () 2015151212200 abxbcycaxy −+−+−×= ()()() () 2015151212200 abxbcycaxy −+−+−×= then the triangle ABC is (1) an acute angle triangle (2) an obtuse angle triangle (3) a right angle triangle (4) an isosceles triangle
67. If , , ijk are unit orthonormal vectors and a is a vector of magnitude 2 units satisfying , then . aijai ×==
(1) 3 ± (2) 2 ± (3) 0 (4) 1
12: Vector Algebra
68. The distance of the point B with P.V . 23 ijk ++ from the line through A with P.V 422ijk ++ and parallel to the vector 236ijk ++ is (1) 10 (2) 5 (3) 6 (4) 2
69. The torque about the point 2 ijk +− of a force represented by 4 ik + acting through the point 2 ijk −+ is (1) 2138ijk ++ (2) 2138ijk−++ (3) 2138ijk +− (4) 2 ijk ++
70. If the position vectors of the vertices of a ∆ ABC are 32, 23 OAijkOBijk =++=++ and 23 OCijk =++ , then the length of the altitude of ∆ ABC, drawn from A is (1) 3 2 (2) 3 2 (3) 3 2 (4) 3 2
71. A (1, 2, 5), B (5, 7, 9), and C (3, 2, –1), are given three points. A unit vector normal to the plane of the triangle ABC (1) 15165 506 ijk +− (2) 15165 506 ijk−+− (3) 15165 506 ijk−++ (4) 3 ijk ++
72. If the area of the parallelogram whose adjacent sides are 34,24 ijkjk λ is
436 sq.units ( λ≥ 0), then λ + 2 = (1) 2 (2) 4 (3) 1 (4) 3
73. The vector equation of the line passing through the point 2 ijk −+ and perpendicular to the vectors 23, 42 ijkijk −−+− is (1) ()() 210311 rijktijk =−++++ (2) ()() 210311 rijktijk =+−+++ (3) ()() 210311 rijktijk =+−+−+ (4) ()() 210311 rijktijk =+−+−−
74. , 102, OAaOBabOCb ==+= where O,A, C are non-collinear points. Let ‘ p ’ denote area of quadrilateral OABC, ‘q’ denote area of parallelogram with OA,OC as adjacent sides then p q = ______.
75. The area of the parallelogram whose diagonals are respectively 32ijk +− and 34 ijk −+ is _____.
76. For any interior point P inside ∆ ABC such that 230PAPBPC++=
then area area ABC APC
is equal to ______.
77. Let
ˆ ˆ ˆ , 232. aijkbijk =+−=+−
Then the vector x satisfying axab ×=× and .0ax = is of length m. Then 2m2 = ____.
78. Let ,, abc be three non-coplanar vectors such that 4 abc ×= , 9 bca ×= and cab ×=α , α > 0. If 36 abc++= , then α = _____.
79. Let ,, abc be unit vectors such that a is perpendicular to the plane of b and c . If (),600bc = then abac×−×= _______.
Multiple Product of Vectors
Single Option Correct MCQs
80. If ···0 xaxbxc=== for some non–zero vector x and 1 abc=== then the volume of parallelopiped with , , abc as coteriminous edges is (1) 1 (2) 2 (3) 3 (4) 0
81. The volume of parallelopiped with edges ()() () 221, OAijk λλλ =+++++
()() ()323 OBijk λλλ =+++++ and ()() () 434, OCijk λλλ =+++++ is (1) 1 (2) 2 (3) 3 (4) 4
82. The value of a so that the volume of parallelopiped formed by , , iajkjak +++ and aik + becomes minimum is (1) –3 (2) 3 (3) 1 3 (4) 3
83. ()()() xabybczca α =×+×+× and 1 8 abc = then x+y + z =
(1) () 8. abcα++ (2) () . abcα++ (3) () 8. abc ++ (4) abc ++
84. The perpendicular distance from origin to the plane passing through the points 22,32,32 ijkijkijk −++−−− is (1) 12 30 (2) 25 110 (3) 10 60 (4) 15 187
86. The shortest distance between the skew lines ()() 2 33 2 rijktijk =+++++ and ()() 45 623 rijktijk =+++++ is (1) 6 (2) 3 (3) 23 (4) 3
87. The vectors ()()() , , abcbcacab ×××××× are (1) coplanar (2) non-coplanar (3) collinear (4) non-collinear
88. ()()() iabijabjkabk ×××+×××+×××= ()()() iabijabjkabk
(1) 0 (2) () . abb (3) b (4) () 2 ab ×
89. Let 22, aijkbij =+−=+ . If c is a vector such that ., 22 accca=−= and the angle b etween () ab × and c is 30°, th en ()abc××=
(1) 2 3 (2) 3 2 (3) 2 (4) 3
90. Let a and b be two non–collinear unit vectors. If () . uaabb =− and ,abν=× . Then ν= (1) u (2) a (3) b (4) ab
91. If ,, abc are non-coplanar unit vectors such that () 2 abcbc + ××= , then the angle between a and b is (1) 4 π (2) 2 π (3) π (4) 3 4 π
85. The shortest distance between the lines whose equations are ()() , 23 rtijkrksijk =++=+−+ is (1) 3 (2) 3 38 (3) 3 14 (4) 2 13
92. If 234, , 423, aijkbijkcijk =+−=++=++ 234, , 423, aijkbijkcijk =+−=++=++ then ()abc××=
(1) 10 (2) 1 (3) 2 (4) 5
93. If ()() abcabc ××=×× where , , abc are any three vectors such that .0, .0abbc≠≠ then a and c are
(1) Parallel
(2) Inclined at an angle of 3 π between them
(3) Inclined at an angle of 6 π between them
(4) Perpendicular
94. 111 · · · abbccc++= (1) 0 (2) 1 (3) 2 (4) 3
95. 111 aabbcc ×+×+×= (1) 0 (2) a (3) b (4) c
96. 111abcabc =
(1) 0 (2) 1 (3) 2 (4) 3
Numerical Value Questions
97. If ,, abc form a left handed orthogonal system and 4,9,16aabbcc ⋅=⋅=⋅= then abc = _____.
98. Let c be a vector perpendicular to the vectors and 2. ˆˆˆˆˆˆ aijkbijk =+−=++ If () .38 ˆ ˆ ˆ cijk++= , then the value of ()cab × is equal to ____.
99. Let three vectors , and abc be such that c is coplanar with a and ,7bac⋅= and b is perpendicular to c , where ˆ ˆ ˆ aijk =−++ and , ˆ ˆ 2 bik =−+ then the value of 2||2 abc ++ is _____.
100. If the volume of the tetrahedron whose vertices are (1, –6, 10), (1, –3, 7), (5, –1, λ) and (7, –4, 7), is 11 cubic units, then the sum of the values of λ, is _____.
101. , , abc are non–coplanar and , , bccaablmn abcabcabc ××× === then ()() . abclmn ++++= _____.
Level – II
Algebra of Vectors
Single Option Correct MCQs
1. The points with position vectors , abab +− and akb + are collinear (1) for exactly two values of k (2) for exactly three values of k (3) for no real value of k (4) for all real values of k
2. If , Pijkqijk λ =++=++
and PqPq +=+
then λ = (1) –1 (2) 1 (3) 2 (4) –2
3. If a αβγδ ++=
and , b βγδαα ++=
and δ are non-collinear, then αβγδ +++ equals (1) aα (2) bδ (3) O (4) () ab+γ
4. Let , ab be two non–collinear vectors. If ()() 421 OAxyaxyb =++++ and ()() 22231 OByxaxyb =−++−− ,
960 OAOB−== O then (x,y) = (1) (1, 2) (2) (1, –2) (3) (2, –1) (4) (–2, –1)
5. Let , ab be two non collinear vectors. If ()() 421 OAxyaxyb =++++ , ()() 22231 OByxaxyb =−++−− and 32OAOB = then (x,y) = (1) (1, 2) (2) (1, –2) (3) (2, –1) (4) (–2, –1)
Numerical Value Questions
6. Let () 2 ab αλ=−+ and ()423ab βλ=−+
be two given vectors where vectors a and b are non– collinear. The value of |λ| for which vectors α and β are co llin ear, is _____.
7. The vectors 2 3 , 5 6 ijij ++ and 8ij +λ have their initial points at (1, 1). The value of λ so that the vectors terminate on one straight line is _____.
8. Let (){} 1 fttittjtk =+−++ where [.] denotes the greatest integer functions. If the vector 5 4 f and ijk λµ++ are parallel then 8λ + μ is _____.
Vectors in Rectangular Cartesian Coordinate System
Single Option Correct MCQs
9. P,Q,R,S have position vectors ,,, pqrs respectively such that ()() 2,pqsr −=− then QS and PR
(1) Bisect each other (2) Trisect each other (3) Parallel to each other (4) all the above
10. The vectors () n ˆ ˆ cossi axxixj =+ and () s ˆ in ˆ bxxixj =+ are collinear for
(1) unique value of ,0 6 xx π <<
(2) unique value of , 63 xx ππ <<
(3) no value of x
(4) infinitely many values of ,0 2 xx π <<
11. The P.V.’s of the vertices of a ∆ ABC are ,4,45. ijkijkijk ++++++ The position vector of the circumcenter of ∆ABC is
(1) 5 3 2 ijk ++ (2) 3 5 2 ijk ++
(3) 1 53 2 ijk ++ (4) ijk ++
12. The points A(2 – x, 2, 2), B(2, 2 – y, 2), C(2, 2, 2 – z) and D(1, 1, 1) are coplanar, then locus of P(x,y,z) is (1) 111 1 xyz ++=
(2) x + y + z = 1
(3) 111 1 111 xyz ++=
(4) none of these
13. If 4 ˆ 7 ˆ 4 ˆ aijk =−− and ˆ ˆ 22 ˆ bijk =−−+ then vector c along the bisector of a and b so that 56 c = is
(1) () ˆ 5 ˆ 2 ˆ 7 3 ijk −+ (2) () ˆ 5 ˆ 2 ˆ 7 3 ijk ++
(3) () 5 7 ˆ ˆ 2 ˆ 3 ijk (4) ˆ ˆ ˆ ijk −+
14. The median AD of the ∆ ABC is bisector at E,BE meets AC in F, then AF : FC is equal to (1) 3 4 (2) 1 2 (3) 1 3 (4) 1 4
15. Let a,b and c be distinct non–negative numbers. If the vectors ai+aj+ck,i+k and ci+cj+bk lie in a plane, then c is:
(1) the harmonic mean of ‘ a’ and ‘b’ (2) is equal to zero (3) the arithmetic mean of ‘ a’ and ‘b’ (4) the geometric mean of ‘ a’ and ‘b’
16. The position vectors of the vertices of a triangle are 345,7 ijkik +++ and 55ij + the distance between the circumcentre and orthocentre is
(1) 0 (2) 306
(3) 2306 (4) 3 306 2
17. If , and abc are non–coplanar vectors and if d is such that ()() 11 and dabcabcd xy =++=++ where x and y are non zero real numbers, then () 1 abcd xy +++=
(1) –a (2) 0
(3) 2a (4) 3a
18. If L1 is a line through the point 5811 ijk ++ and parallel to the vector 234ijk ++ and L2 is a line through the point 46 8 ijk ++ and parallel to the vector 345ijk ++ , then the point of intersection of L1 and L2 is (1) ijk ++ (2) 2 3 ijk ++ (3) 2 3 ijk ++ (4) 2 2 ijk −+
Numerical Value Questions
19. In ∆ABC, P,Q,R are points on ,, BCCAAB respectively, dividing them in the ratio 1 : 4, 3 : 2 and 3 : 7. The point S divides AB in the
ratio 1 : 3. Then 5 APBQCR CS ++ = _____.
20. The points represented by ,,, abcd are coplanar and ()()() sin2sin23sin340 AaBbCcd ++−=
then, the least value of 21 8 (sin2A + sin22B + sin23C) is _____.
21. D,E,F are respectively the points on the sides BC , CA , and AB of ∆ ABC dividing them in the ratio 2 : 3, 1 : 2, 3 : 1 internally. The lines BE and CF interset on the line AD at p. If 11 .. APxAByAC =+ then x1 + y1 = _____.
Scalar Product of Two Vectors
Single Option Correct MCQs
22. If ,and abc are unit vectors, then 22 2 abbcca −+−+− does not exceed (1) 4 (2) 9 (3) 8 (4) 6
23. If abab +<− then the angle between a and b can lie in the interval (1) 0, 2 π (2) , 2 π π
(3) 0, 4 π (4) , 42 ππ
24. If ,, xyz are three unit vectors in three dimensional space, then the minimum value of 222 xyyzzx +++++ is (1) 3 2 (2) 3
(3) 33 (4) 6
25. In ∆ABC, CBa = , CAb = , ABc = CD is median through the vertex C. Then CACD equals
(1) () 222 1 3 4 abc +− (2) () 222 1 3 4 abc +− (3) () 222 1 3 4 abc +− (4) () 222 1 3 4 abc−++
26. If a,b,c are p th , q th , r th terms of an H.P and ()()() , uqrirpjpqk =−+−+− , j ik v abc =++ then
(1) , uv are orthogonal vectors (2) , uv are parallel (3) can not be determined (4) none of these
27. If the pth , qth , rth terms of a GP are the positive numbers a,b,c then the angle between the vectors ()()() 222 log log log aibjck ++ and ()()() qrirpjpqk −+−+− is
(1) 4 π (2) 2 π (3) 6 π (4) π
28. O is the origin in the cartesian plane. From the origin ‘O’ take point A in the North East direction such that 5. OA = B is a point in the NorthWest direction such that 5. OB = Then OAOB⋅=
(1) 25 (2) 52 (3) 105 (4) 5
29. If A,B,C,D are four points in space satisfying 2222 . ABCDkADBCACBD =+−−
then k = (1) 2 (2) 1 2 (3) 1 2 (4) 2
30. If the vector 34 ajk =+ is the sum of two vectors 1a and 2a , vector 1a is parallel to bij =+ and vector 2a is perpendicular to ,b then 1a = (1) () 1 2 ij + (2) () 1 3 ij + (3) () 2 3 ij + (4) () 3 2 ij +
Numerical Value Questions
31. If ,, abc are unit vectors satisfying 22 2 9 abbcca −+−+−= then 355 abc ++ is _____.
32. For p > 0, a vector () 2 2 ˆ ˆ 1 vipj =++ is obtained by rotating the vector 1 ˆ 3 ˆ vpij =+ by an angle q about origin in co un ter clockwise direction. If tan () () 32 , 433 α θ= + then the value of α is equal to _____.
33. Let G be the centroid of the triangle ABC whose sides are of length a,b,c if ‘P’ is point in the plane of triangle ABC such that PA = 1, PB = 3, PC = 2, PG = 2, then (a2 + b2 + c2) = _____.
34. Vectors along the adjacent sides of a parallelogram are 2 ˆ ˆ ˆ aijk =++ and 4 ˆ 2 ˆ ˆ bijk =++ then, the length of the longer diagonal of the parallelogram is ____.
35. If a and b are any two unit vectors, then the minimum value of 22 11 |||| abab + +− is _____.
Cross Product of Two Vectors
Single Option Correct MCQs
36. Let , ab and c be three non–coplanar vectors and d be a non–zero vector, which is ⊥ to () abc ++ . Now if ()()()sincos2 dxabybcca =×+×+×
then minimum value of x2 + y2 is equal to (1) π2 (2) 0 (3) 2 4 π (4) 2 5 4 π
37. If a and b are two vectors of magnitudes 2 and 3, respectively, such that ()() 23.ababk ×+= then the maximum value of k is
(1) 13 (2) 213 (3) 613 (4) 1013
38. If , , abc are the position vectors of A,B,C of ∆ ABC, then ()()() abbcca ×+×+×=
(1) 1 2 (Area of ∆ ABC) (2) 2(Area of ∆ ABC) (3) 3(Area of ∆ ABC) (4) none
39. If a and b are vectors such that 29 ab+=
40. If A1, A2,….A n are the vertices of a regular polygon wigh ‘n ’ sides and O is its centre then 1 1 1 n ii i OAOA + = ∑×=
(1) () () 12 1 nOAOA −×
(2) () () 12 1 nOAOA +×
(3) () 12 nOAOA × (4) 1 2OAOA ×
41. Let , ab and c be three unit vectors such that 0. abc++= .If abbcca µ=×+×+× and abbccaλ+ =+ then the ordered pair () , λµ is
×
(1) () 3 ,3 2 ab × (2) () 3 ,3 2 cb
(3) () 3 ,3 2 bc ×
(4) () 3 ,3 2 ac
×
42. , pq and r are three mutually perpendicular vectors of the same magnitude. If vector x satisfies the equation () ()()() pxqpqxrqr ×−×+×−×+×
() () 0 xpr−×= ,then x is given by (1) () 1 2 2 pqr +− (2) () 1 2 pqr ++
(3) () 1 3 pqr ++
(4) () 1 2 3 pqr +−
aijkijkb ×++=++×
and ()() 234234,
then the absolute value of ()() ˆ . ˆ 3 ˆ 72 abijk +−++ is (1) 0 (2) 3 (3) 4 (4) 8
43. The absolute value of ‘ t ’ for which area of ∆ formed by A (–1, 1, 2), B (1, 2, 3) and C(t, 1, 1) is minimum _____.
44. Let the vectors ,, abc and d be such that ()() 0 abcd×××= and P 2 be planes determined by the pairs of vectors , ab and , cd
respectively. Then the angle between P1 and P2 in degrees is _____.
45. If ,, abc are three vectors such that 1,4,2,2 acbbcbca ===×==+λ where λ is a scalar. If the value of ‘λ’ is equal to 3 αβ , where α, β are real numbers, then the value of α + β = ______.
51. ()()()()()() abbcbccacaab ×××××××××=
()()()()()() abbcbccacaab ×××××××××=
(1) abc (2) 2 abc
(3) 3 abc (4) 4 abc
46. Let , and abc be unit vectors such that 0 abc+−= . If the area of triangle formed by vectors a and b is A, then what is the value of 16A2 ______?
47. Let a and b be two vectors such that 222 ||2||,3ababab +=+⋅= and
2 ||75. ab×= Then ||2 a is equal to ______.
48. If A,B,C and D are four points in space, then ABCDBCADCABDk ×+×+×= (area of ∆ABC), where k is equal to ______
Multiple Product of Vectors
Single Option Correct MCQs
49. ()() , 1 , 1 aikbxijxkcyixjxyk =−=++−=+++−
()() 1 , 1 aikbxijxkcyixjxyk =−=++−=+++− then abc depends on (1) only x (2) only y (3) neither x nor y (4) both x and y
50. If V is the volume of the parallelopiped having three coterminous edges as , ab and c then the volume of the parallopiped having three coterminous edges as ()() .(.). aaaabbaccα=++
()()() ... baabbbbccβ=++
()()() ... caacbbcccγ=++
(1) V (2) V2 (3) V3 (4) V4
52. The condition that the lines , ratbrcsd =+=+ to intersect each other is (1) [ |0 cabd−= (2) [ |0 bdac−= (3) [ |0 abcd−= (4) [ |0 cdab−=
53. Let , and abc be three nonzero vectors such that no two of them are collinear and () 1 3 abcbca ××= . If θ is the angle bet ween ve ctors and bc , t hen a value of sinθ is (1)
Numerical Value Questions
54. The lengths of two opposite edges of tetrahedron are a and b and their shortest distance is d and angle between them is q. If volume of tetrahedron is 1 sin abd k θ then k is ______.
55. If , , abc are mutually perpendicular unit vectors then
56. Let ,, abc be coplanar unit vectors such that cos ,cos ,cosbccaabαβγ⋅=⋅=⋅= then the value of cos 2α + cos 2 β + cos 2 γ –2cosαcos β cos γ is _____.
57. Let 1,1ab== and 3 ab+= . If c be a vector such that () 23 cabab =+−× and () , pabc =×× then p 2 is equal to _____.
58. b and c are non collinear vectors, if ()() ()() 2 . 42sin1 aacabb xybxc ××+= −−+− and () . ccac = then the value of xsiny + (4siny)4x + 15 must be ______.
Level – III
Single Option Correct MCQs
1. Let S be the set of all (λ, μ) for which the vectors ,2 ijkijk λµ −+++ and 345ijk −+ , where λ – μ = 5 are coplanar then () () 22 , 80 s λµ λµ ∈ ∑+ is
(1) 2370 (2) 2130 (3) 2210 (4) 2290
2. If b is a vector whose initial point divides the join of 5 an ˆ ˆ d 5 ij in the ratio k : 1 and whose terminal point is the origin and 37 b ≤ , then k lies in the interval
(1) 1 6, 6 (2) 1 ,6, 6
(3) [0, 6] (4) (–6, 0)
3. If 35 ijk ∧∧∧ −+ bisects the angle between a ∧ and 22 ijk ∧∧∧ −++ where a ∧ is a unit vector, then (1) 1 418840 105 aijk
(2) 1 418840 105 aijk
(3) 1 418840 105 aijk
(4) 1 418840 105 aijk
4. Let two non–collinear unit vectors ˆ ˆ and ab form an acute angle. A point P mov es so that a t any time “ t ”, the position vector OP (where ‘ O ’ is the origin) is given by n ˆ ˆ cossi atbt + . When ‘P’ is farthest from origin ‘ O ’ let ‘ M ‘ be the lengt h of OP and ˆ µ be the unit vector along OP , then (1) () 1 2 a
nd
ˆ 1. ab Mab ab µ==+ (2) () 1 2 and12
ˆ ab Mab ab µ + ==+ + (3) () 1 2 a
nd
1. ab Mab ab µ + ==+ + (4) () 1 2 and12
Mab ab µ==+
5. If the points with position vectors 9 1013,61111, ˆˆˆˆˆˆˆˆˆ 8 2 ijkijkijk αβ +++++− are collinear, then (19α – 6 β )2 is equal to (1) 16 (2) 36 (3) 25 (4) 49
6. The area of quadrilateral ABCD with vertices A (2, 1, 1), B (1, 2, 5), C (–2, –3, 5) and D(1, –6, –7) is equal to (1) 9 38 (2) 48 (3) 54 (4) 838
7. Let the vectors 2 1 , ˆˆˆˆˆˆ uijakuibjk =++=++
and 3 ˆ ˆ ˆ ucijk =++ . If the vectors
, vabicjckvaibcjak =+++=+++
and () 3 ˆ ˆ ˆ vbibjcak =+++ are also coplanar, then 6(a + b+c) is equal to (1) 12 (2) 4 (3) 0 (4) 6
8. Let O be the origin and the position vector of the point P be ˆ ˆ 3 ˆ 2 ijk−−+ . If the position vectors of the points A, B and C are 23,242 and 42
, ijkijkijk −+−+−−+− respectively, then the projection of the vector OP on a vector perpendicular to the vectors and ABAC is (1) 8 3 (2) 3 (3) 10 3 (4) 7 3
10. Let ˆ 27, ˆ 35 ˆ ˆ ˆ aijkbik =+−=+ and ˆ ˆ ˆ 2 cijk =−+ Let d be a vector which is perpendicular to both a and b and 12 cd⋅= . Then ()() ˆ ˆ ˆ ijkcd −+−⋅× is equal to (1) 24 (2) 44 (3) 48 (4) 42
11. If the points P and Q are respectively the circumcenter and the orthocenter of a ∆ABC, then PAPBPC ++
is (1) 2 PQ
(3) PQ
(2) 2QP
(4) QP
12. Let a be a non-zero vector parallel to the line of intersection of the two planes described by , ˆ ˆ ˆˆ ijik ++ and , ˆ ˆˆ ˆ ijjk .If q is the angle between the vector a and the vector ˆ 2 ˆ 2 ˆ bijk =−+ and .6ab = , then the ordered pair () , abθ× is equal to
(1) ,6 3
(3) ,36 3 π
(2) ,6 4
(4) ,36 4
13. For any vector 123 ˆ ˆ ˆ aaiajak =++ , with 10 | ai| < 1, i = 1, 2, 3, consider the following statements:
and , OQuv αβ=+
9. An arc PQ of a circle subtends a right angle at its centre O. The Midpoint of the arc PQ is R. If , OPuORv ==
then α, β 2 are the roots of the equation
(1) x2 + x – 2 = 0
(2) x2 – x – 2 = 0
(3) 3x2 + 2x – 1 = 0
(4) 3x2 – 2x – 1 = 0
(A) : {}max,,123aaaa ≤
(B) : {}3max,,123 aaaa ≤
(1) Neither (A) nor (B) is true
(2) Only (A) is true
(3) Only (A) is true
(4) Both (A) and (B) are true
14. If four distinct points with position vectors ,, abc
and d are coplanar, then abc
is equal to
15. Let , Zaijkλλ ∧∧∧ ∈=+−
and 32. bijk
Let c
If δ > 0 and the area of the triangle ABC is 5 6 , then CBCA ⋅ is equal to (1) 108 (2) 120 (3) 54 (4) 60
19. Let 2,3ab== and the angle between the vectors and be 4 ab π . Then
()() |223|2 abab +×− is equal to (1) 441 (2) 882 (3) 841 (4) 482
20. Let ABCD be a quadrilateral. If E and F are the Midpoints of the diagonal AC and BD respectively and
be a vector such that () 0,.17abccac ++×==−
Then 2 cijk λ ∧∧∧
×++
and .20bc =−
is equal to (1) 53 (2) 49 (3) 46 (4) 62
16. Let a,b,c be three distinct real numbers, none equal to one. If the vectors , aijkibjk ∧∧∧∧∧∧ ++++ and ijck ∧∧∧ ++ are coplanar, then 111 111abc ++ is equal to (1) –2 (2) –1 (3) 1 (4) 2
17. Let 42, 32
7 ˆˆˆˆ aijkbijk =++=−+
and 4 ˆ ˆ 2 ˆ cijk =−+ . If a vector d satisfies dbcb ×=×
and .24,da = then 2 d is equal to (1) 423 (2) 313 (3) 413 (4) 323
18. Let for a triangle ABC, 3
2
43 ABijk CBijk CAijk αβγ δ =−++ =++ =++
()() ABBCADDCkFE −+−= then k = (1) –4 (2) –2 (3) 2 (4) 4
21. Let S be the set of all (λ, μ) for which the vectors λ i – j + k , i + 2 j + μk and 3 4 5 ijk −+ where λ – μ = 5 are coplanar then () 22 (, ) 80 S λµ λµ ∈ ∑+ is (1) 2290 (2) 2210 (3) 2370 (4) 2130
22. Let 2,2,2 ˆˆˆˆˆˆ uijkvijkvw =−−=+−⋅= and vwuv ×=+λ .Then uw ⋅ is equal to (1) 1 (2) 2 (3) 3 2 (4) 2 3
23. Let 435 and 24 . ˆ ˆˆˆˆ ijkijkαβ=++=+− Let 1β be parallel to 2 and αβ → be perpendicul ar to α . If 12 βββ =+ , then the value of () 2 5 ˆ ˆ ˆ ijkβ++ is (1) 11 (2) 6 (3) 9 (4) 7
24. The vector ˆ ˆ ˆ 2 aijk =−++ is rotated through a right angle, passing through the y–axis in its way and the resulting vector is b . Then the
projection of 32ab + on 43 ˆ 5 ˆ ˆ cijk =++ is (1) 32 (2) 1 (3) 23 (4) 6
Numerical Value Questions
25. ABC is a triangle and ‘O’, any point in the plane of the triangle. The lines AO,BO and CO meet the sides BC,CA and AB in D,E, F respectively.
Then ODOEOF ADBECF ++ equal to _____.
26. Let Q and R be two points on the line 2 11 232 y xz + +− == at a dist ance 26 from the point P(4, 2, 7).
Then the square of the area of the triangle PQR is ____
27. Let , , abc be three non–coplanar vectors such that 4, 9 abcbca ×=×=
and , 0 cabαα ×=>
. If 36 abc++=
, then α is equal to _____.
28. Let a and b be two vectors such that 22 2 2,.3ababab +=+=
and 2 75 ab×=
Then 2 a is equal to_____.
29. Let 23, ad
aijkbijkc =−+=++
n
be a vector such that () 0 and 5 abcbc +×=⋅=
. Then, the value of ()3.ca is equal to _____.
31. Let , and abc be three unit vectors such that 22 ||8.abac−+−= Then 22 2|2| abac +++ is equal to ______.
32. Let P be a plane passing through the points (1, 0, 1), (1, −2, 1) and (0, 1, −2). Let a vector ˆ ˆ ˆ aijk αβγ=++ be such that a is parallel to the plane P, perpendicular to () ˆ 3 ˆ 2 ˆ ijk ++ and () .22 ˆ ˆ ˆ aijk++= , then ()2 αβγ −+ equals _____.
33. For p > 0, a vector () 2 1 ˆ ˆ 2 Vipj =++ is obtained by rotating the vector 1 ˆ 3 ˆ vpij =+ by a n a ngle q about origin in counter clockwise direction. If () () 32 tan 433 α θ= + , then the value of α is equal to ______.
34. If the projection of the vector ˆ ˆ ˆ 2 ijk ++ on the sum of the two vectors ˆ 25 ˆ 4 ˆ ijk +− and ˆ ˆ ˆ 23 ijkλ −++ is 1, then λ is equal to ______.
30. If 123 23,33 ˆ ˆˆˆˆˆ and ˆˆˆ aijkbijkccicjck =++=++=++
and
123 23,33
are coplanar vectors and 5, acbc ⋅=⊥ , then 122(c1 + c2 + c3) is equal to ______.
35. Let 5, 3 ˆˆˆˆˆˆ aijkbijk αβ=++=++ and ˆ ˆ ˆ 23 cijk =−+− be three vectors such that, 53 bc×= and a is perpendicular to b . The n, the greatest amongst the values of 2 a is ______.
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. For what value of p , paa < and 1 2 paa + is parallel to a ?
2. If ()kab + is unit vector then what is the value of k ?
3. If , ab are any two non-zero vectors and having equal magnitude then what is the vector which bisects te angle between them?
4. If ˆ ˆ ˆ 23 aijk =+− and ˆ ˆ ˆ 249 bijk =++ , then find the unit vector parallel to ab +
5. If ˆ ˆ ˆ 23 aijk =+− and ˆ ˆ ˆ 249 bijk =++ , then find the unit vector perpendicular to both the vectors.
6. Find the value of p such that ˆ ˆ ˆ 329ijk ++ and ˆ ˆ ˆ 23 ipjk −+ are parallel to each other.
7. If ˆ ˆ ˆ 2 axijzk =+− and ˆ ˆ ˆ 3 biyjk =−+ are equal, then find the value of xyz ++
8. Find a vector a of magnitude 52 making an angle 4 π with x axis, 2 π with z axis and obtuse angle with y axis.
9. If 4 a = , then find the range of ka where 32 k −≤≤
10. Write the position vector of a point which divides the join of points with position vectors 32ab and 23ab + in the ratio 2:1
Statement Type Questions
Each question has two statements: statement I (S-I)and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct
(2) if both statement I and statement II are incorrect
(3) if statement I is correct, but statement II is incorrect
(4) if statement I is incorrect, but statement II is correct
11. S – I : The angle between the vectors ˆ ˆ ˆ aijk =−+ and ˆ ˆ ˆ bijk =+− is obtuse angle.
S - II : If ab ⋅ is negative then the angle between a and b is obtuse.
12. S - I : If two vectors ˆ ˆ ˆ 329ijk ++ and ˆ ˆ ˆ 3 ipjk ++ are parallel vectors, then 2 3 p =
S - II : If two vectors 123 ˆ ˆ ˆ aaiajak =++ and 123 ˆ ˆ ˆ bbibjbk =++ are parallel then 3 12 123 . aaa bbb ==
13. S - I : If ,, abc are three mutually perpendicular unit vectors, then 3. abc++=
S - II : ()()() 2222 222 abcabc abbcca ++=++ +⋅+⋅+⋅
14. S - I : The projection of bc + on a where () 2,2,1 a , () 1,2,2 b and () 2,1,4 c =− is 2
S - II : The projection of bc + on a is baca a ⋅+⋅
15. S - I : If a and b are two unit vectors such that 2332 abab +=− , then the angle between a and b is right angle.
S - II : The magnitude of each of the two vectors a and b for which () , 3 ab π = and 9 2 ab⋅= is 3
16. S - I : If ˆ ˆ ˆ 2 aiyjk =++ and ˆ ˆ ˆ 23 bijk =++ are two vectors for which the vector ab + is perpendicular to ab when 3 y =±
S - II : If ab + is perpendicular to ab then magnitudes of both , ab are equal.
17. S - I : If a and b are unit vectors and θ is angle between them then sin 2 ab θ −=
S - II : If a and b are unit vectors and θ is angle between them, then cos. 2 ab θ +=
18. S - I : If the difference of two unit vectors is a unit vector then the angle between them is 60. °
S - II : If the sum of unit vectors is unit vector then the angle between them is 2 . 3 π
19. S - I : If the sum of two unit vectors is a unit vector, the magnitude of their difference is 3.
S - II: If the difference of two unit vectors is a unit vector, the magnitude of their sum is 2
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
(4) if both (A) and (R) are false
20. (A) : The angle between the vectors a and b where 3 a = , 2 b = and 6 ab⋅= is . 4 π
(R) : If θ is angle between two vectors a and b then cos. ab ab ⋅ θ=
21. (A) : If ˆ ˆ ˆ 2ijk +λ+ and ˆ ˆ ˆ 23 ijk −+ are orthogonal to each other then [ λ ] =3 where [.]represents greatest integer function.
(R) : If two vectors 123 ˆ ˆ ˆ aaiajak =++ and 123 ˆ ˆ ˆ bbibjbk =++ are parallel then 112233 0 ababab++=
22. (A) : The projection of ˆ ˆ ˆ 37 ijk ++ on the vector ˆ ˆ ˆ 236ijk −+ is 5.
(R) : The projection of a on b is . ab b ⋅
23. (A) : If , ab are two vectors such that 2 3, 3 ab== and ab × is a unit vector then the angle between a and b is 30. °
(R) : If θ is acute angle between a and b then sin. ab ab × θ=
24. (A) : If 10 a = , 2 b = and 12 ab⋅= , then the value of 16. ab×=
JEE ADVANCED LEVEL
Multiple Option Correct MCQs
1. The sides of a parallelogram are ˆ 25 ˆ 4 ˆ ijk +− and ˆ 3 ˆ 2 ˆ ijk ++ . The unit vector parallel to one of the diagonals is/are
(1) () 1 36 ˆ 2 ˆ 7 ˆ ijk +−
(2) () 1 36
ijk
(3) () 1 2 ˆ 8 69 ˆ ˆ ijk ++ (4) () 1 2 ˆ 8 69 ˆ ˆ ijk−−+
2. If the side AB of an equilateral triangle ABC lying in the xy plane is 3 ˆ i , then side CB can be
(1) () ˆ 3 2 ˆ 3 ij (2) () 3 2 ˆ 3 ˆ ij (3) ()
(R) : 222 2 . ababab ×+⋅=
25. (A) : If 0 rarbrc⋅=⋅=⋅= for some nonzero vectors ,, abc , then the value of ()abc ⋅× is zero.
(R) : If ,, abc are coplanar vectors, then 0. abc
3. The vector 2 ˆ ˆ 5 ˆ axijk =−+ and ˆ ˆ ˆ biyjzk =+− are collinear, if
(1) x = 1, y = –2, z = –5
(2) x = 1 2 , y = –4, z = –10
(3) x = 1 2 , y = 4, z = 10
(4) x = –1, y = 2, z = 5
4. If A(–4, 0, 3) and B(14, 2, –5) then which one of the following points lie on the bisector of the angle between and OAOB
(O is the origin of reference)?
(1) (2, 2, 4) (2) (2, 11, 5)
(3) (–3, –3, –6) (4) (1, 1, 2)
5. The vectors ()()() ˆˆ ,3 ˆ 12 ˆ xixjxkxi ++++++
()() 5 ˆ 4 ˆ xjxk +++ and ()()() 7 ˆ 6 ˆ 8 ˆ xixjxk +++++ are coplanar if x is equal to (1) 1 (2) –3 (3) 4 (4) 0
6. A vector of magnitude 2 along a bisector of the angle between the two vectors
22ijk −+ and 22 ijk +− is
(1) () 2 3 10 ik
(2) () 1 43 26 ijk −+
(3) () 2 43 26 ijk −+
(4) () 1 10 ˆ 3 ˆ ik +
7. If the vectors (–bc, b2 + bc, c2 + bc), (a2 + ac, –ac, c2 + ac) and (a2 + ab, b2 + ab, –ab) are coplanar, where none of a,b or c is zero then
(1) a2 + b2 + c2 = 1
(2) a + b + c = 0
(3) ab+bc + ca =0
(4) a2 + b2 + c2 = (a + b + c)2
8. All values of λ such that x,y,z ≠(0,0,0) and ()()()
10. The position vectors of the vertices A,B,C of a tetrahedron ABCD are , ijki ++ and 3i respectively. The altitude from the vertex D to the opposite face ABC meet the median line through A of the ∆ABC at E. If the length of side AD is 4 and volume of tetrahedron is 22 3 then P.V of E is
(1) 33 ijk−++ (2) 3ijk (3) 3 ijk −+ (4) ijk
11. Let , and xyz be three vectors each of magnitude 2 and the angle between each pair of them is 3 π . If a is a non–zero vector perpendicular to and and xyzb × is a non zero vector perpendicular to and yzx × , then
33345
ijkxijkyijz +++−++−+
ˆ ˆ xiyjzk λ =++ whe re ˆ ,,
ˆ ijk are unit vectors along the coordinate axes. (1) 0 (2) 1 (3) –1 (4) 2
9. Let 2,2
be three vectors. A vector in the plane of and bc whose p roj ection on a is of magnitude 2 3 is (1) ˆ 23 ˆ 3 ˆ ijk +− (2) ˆ 23 ˆ 3 ˆ ijk ++ (3) ˆ 2 ˆ 5 ˆ ijk−−+ (4) 2 ˆ 5
ˆ ijk ++
(1) () ()bbzzx =⋅−
(2) () ()aayyz =⋅−
(3) () ()abaybz ⋅=−⋅⋅ (4) () ()aayzy =⋅−
12. In triangle ABC, uv AB uv =− and 2u AC u = , where , uv ≠ then
(1) 1 + cos2A + cos2B + cos2C = 0 (2) sinA ≡ cosC (3) Projetion of AC on BC is equal BC (4) Projetion of AB on BC is equal to AB
13. If and ab are non-zero vectors such that 2 abab +=− then
(1) 2 2.abb = (2) 2 abb =
(3) least value of 2 1 .is2 2 ab b + + (4) least value of 2 1 .is21 ||2 ab b +− +
14. If OABC is a tetrahedron such that OA 2 + BC2 = OB2 + CA2 = OC2 + AB2, then (1) OA ⊥ BC (2) OB ⊥ CA (3) OC ⊥ AB (4) AB ⊥ BC
15. If 230abc++= then abbcca ×+×+×= (1) 0 (2) () 2 ab ×
(3) () 3 ca × (4) () 6 bc ×
16. The angles of a triangle, two of whose sides are represented by vectors
()() ˆ 3an.ˆˆ d abbaba ×− where b is a non zero vector and ˆ a is a unit vector in the direction of a , are (1) () 1 tan3 (2) 1 1 tan 3
(3) 2 π (4) tan–1(1)
17. Let , and abc be three non–coplanar vectors and d be a non–zero vector perpe ndicular to () abc ++ . Now ()()() sincos2 dabxbcyca =×+×+×
then
(1) () 2 dac abc ⋅+ =
(2) () 2 dac abc ⋅+ =−
(3) minimum value of x2 + y2 is
(4) minimum value of x2 + y
18. If each of ,, abc is orthogonal to sum of other two vectors and 3,4,5abc=== then
(1) when a is equally inclined with coordinate axes, then tan2 θ=±
(2) range of ab is [1, 7]
(3) range of bc is [1, 9]
(4) range of ac is [1, 5]
19. Let , and abc be non–zero vectors and ()()12 and VabcVabc =××=××
Vectors
V1 and V2 are equal. Then
(1) and ab are orthogonal
(2) and ac are collinear
(3) and bc are orthogonal (4) ()bac λ =× when λ is scalar
20. If ABCD is a regular tetrahedron with length of any edge be 7, then
(1) volume of tetrahedron is 343 62
(2) volume of tetrahedron is 243 2
(3) minimum distance of any vertex from the opposite face is 2 7 3
(4) minimum distance of any vertex from the opposite face is 3 7 2
21. If the planes 12 .,.2, rijkqriajkq
∧∧∧∧∧∧ ++=++=
and 2 3 . raiajkq
∧∧∧ ++= intersect in a line, then the value of a is (1) 1 (2) 1 2 (3) 2 (4) 0
22. Unit vectors and ab are perpendicular and unit vector c is inclined at an angle q to both and ab , if cabab αβγ = =++×
then
(1) α = β (2) γ 2 = 1 – 2α2 (3) γ 2 =– cos2 q (4) 2 1cos2 2 βθ + =
23. A non-zero vector a is parallel to the line of intersection of the plane determined by the vectors , ˆ ˆˆ iij + and the plane determined by the vecto rs , ˆ ˆ ˆˆ ijik −+ . Then the angle between, a and the vector ˆ 2 ˆ 2 ˆ ijk −+ (1) 45° (2) 30° (3) 135° (4) 60°
24. If and ab be two non collinear unit vectors. If () · uaabb =− and ,vab =× then v = (1) u (2) uua + (3) · uub + (4) () . uuab ++
25. Unit vectors ,, abc are coplanar. A unit vector d is perpendicular t o them. If ()() 111 633 abXcdijk ××=−+ then which of the following vectors are parallel t o c (1) 244ijk −+ (2) 22 ijk−+− (3) 23 ijk−++ (4) 2 ijk−++
26. abcdef ×××
is equal to (1) [][]-[][] abdcefabcdef
(2) [][]-[][] abefcdabfecd
(3) [][]-[][] cdabefcdbaef
(4) [][] acebdf
27. Let OABC be a tetrahedron.The position vectors of A,B,C are
,,
iijjk ++ respectively, O is the origin. Consider the plane ABC as base (1) height of the tetrahedron form origin is 1 2
(2) area of the base triangle ABC is 1 2
(3) volume of tetrahedron is 1 6 (4) volume of tetrahedron is 1 6
28. Let ,and abc be three non-coplanar vectors forming a right-handed system. Let , bcca pq abcabc ×× ==
If x ∈ R+, then (1) xabcpqr x
has least value 2
Numerical/Integer Value Questions
29. If q is the acute angle between the medians drawn through the acute angles of an isosceles right angled triangle then the value of 4sec q = _____.
30. Forces acting on a particle have magnitudes of 5, 3, and 1 units and act in the direction of the vectors 623,326 ijkijk ++−+ and 236ijk respectively. They remain constant while the particle is displaced from the point A (2, –1, –3) to B (5, –1, 1). The work done is equal to k, then 11 k = ______.
31. If 1,2ab== and the angle between a and b is 120° then ()() {}2 33 abab+×−= _____.
32. A straight line L cuts the lines AB,AC and AD of a parallelogram ABCD at points B1, C1 and D1 respectively. If 11 1 123 , , and ABABADADACAC ===
λλλ then 312 111 1
33. Let ,, abc be vectors in three dimensional space where and ab are unit vectors which are not perpendicular to each other and .2,.2,.16. acbccc=== If the volume of the parallelopiped, whose adjacent sides are represented by the vectors ,and abc is 8 , then the integral part of 35ab is _____.
(() () )() ....... k aaaabaab ×××××=−× .
In the left-hand side of the equation we a consider in 2023 times, then | k – 2022 | = _____.
36. If h is the altitude of a parallelepiped determined by the vectors ,, abc and the base is taken to be the parallelogram determined by and ab where ˆ ,24 ˆˆˆˆ ˆ aijkbijk =++=+− and ˆ 3 ˆ ˆ cijk =++ then the value of 19 h2 is ___.
37. If ,and abc are perpendicular to ,and bccaab +++ , respectively, and if 6,8,10abbcca +=+=+= and abc ++ is equal to p then 2 p equal to _____.
38. Let , and abc be three unit vectors such that 22 ||8abac−+−= . Then 22 2|2| abac +++ is equal to _____.
39. A set of ‘ n ‘ vectors 12,,................ n aaa are said to be linearly dependent if there exist ‘ n ‘ scalars x1, x2, x3 x n, not all simultaneously zero’s such that 1122 .....0. nn xaxaxa+++= If 25and axijkbiyjzk ∧∧∧∧∧∧ =−+=+− are linearly dependent, then the value of 2 xy z is 5 k , then the value of k is ______.
34. If ,, abc are three non coplanar, non zero vectors and r is any vector in space such that ()()()()()() abrcbcracarb ×××+×××+××× ()()()() abrcbcracarb ×××+×××+××× abcr λ = then λ is ___.
35. Let , ab non – collinear vectors, .0,ab ≠ and
40. Let q be the angle between the vectors and ab , where 4,3, ,. 43 ab
==∈
Then ()() 22 | |4(·)ababab −×++ is equal to ______.
41. Let ,, abc be three non –coplanar vectors such that 4,9and abcbcacab α ×=×=×= , α > 0. If 36, abc++= , then 5 α is __.
42. If vectors , and abc are such that 1 23 bc a === , then the maximum value of 222 |||| abbcca −+−+− is λ 1 , then number of positive integral divisors for λ1 is λ2 and the number of positive integral even divisors for λ 1 + 2 is λ 3, then 1 23 2λ
([.] denotes GIF ) _____.
43. If and ab are any two perpendicular vectors of equal magnitude such that 344320, then ababa ++−== ______.
44. The value of p exists so that the straight lines ()() ˆˆˆˆˆˆ 291323 rijktijk =+++++ and ()() ˆˆˆˆˆˆ 37 23 rijpksijk =−+++−+− are coplanar. If the point of intersection of the two lines is (α, β , γ ) then the value of α + β + γ – p is equal to ______.
45. Let for a triangle ABC, ; ˆ 3 ˆ 2 ˆ ABijk =−++ ˆˆˆˆˆˆ ; 43 CBijkCAijk αβγδ =++=++ . If δ > 0, and the area of the triangle ABC is 56, then () 1 15 CBCA is equal to ______.
46. If ,, abc are the vectors such that 2 abc = , then the value of abbccaabbcca +++−××× ()()() abcbcacab +×××××× is equal to _____.
47. In a triangle PQR, let , aQRbRP ==
and .cPQ = If 4,5ab== and () ()() .2 , 2 23.2 abca cababab = ++−+
then the value of greatest integer less than or equal to 2 ab ×
is ______.
Passaged-based Questions
Q. (48-49)
Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel. Also, OA : CB = 2 : 1 and OD : AB = 1 : 3 then answer the following questions
48. The ratio OX XC is m n , then (HCF of m,n is 1) n – m = _____.
49. The ratio AXp XDq = (HCF of p,q is 1)then p+q = _____.
Q. (50-51)
Consider the regular hexagon ABCDEF with center at O (origin).
50. ADEBFC ++
is equal to nAB
, then n = _____.
51. Five forces , , , , ABACADAEAF
act at the vertex A of a regular hexagon AB CDE F Then, their resultant is mAO
Q. (52-53)
, then m =
Consider a Parallelogram ABCD with E as the midpoint of its diagonal BD . The Point ‘ E ’ is connected to a point F on DA such that 1 3 DFDA = and to a point G on BC such that 1 4 BGBC =
52. Ratio of Area of Quadrilateral ABEF to the area of triangle DEF be K : 1. Then K = ______.
53. Ratio of Area of quadrilateral AEGB to the area of triangle GEC is λ : 1. Then λ = ______.
Q. (54-55)
Let ,and abc be three non–coplanar unit vector such that the angle between every pair of them is 3 π If , abbcpaqbrc ×+×=++
where p, q, r are scalars; then
54. The value of 4 abc is ______.
55. The value of 20p2 + 21q2 – 22r2 is ______.
Q. (56-57) , UijkVaibjck =++=++ ( V is non-zero vector) (where i,j,k are unit orthogonal vectors). Here a,b,c ∈ {–2, –1, 0, 1, 2}
56. Total number of possible V , such that .0UV = is _____.
57. Total number of possible V , such that 0 UV×= is _____.
Q. (58-59)
If and ab are two vectors, then .cosabab=θ
and sin abab×=θ
, where q is angle between two vectors, then
58. Let ,and abc be three vectors such that 2,3ab== and 5 c = satisfying
30, abc
then ()()() () () 2.abcacacb ++××−+
is equal to _____.
59. Let ,, abc be three vectors satisfying 2 abcb =×+ , where 2 bc== and 4, a ≤ then the sum of all possible values of 2abc ++ is ______.
Q. (60-61)
ABCD is a tetrahedron having each edge of unit length.
60. The volume of the tetrahedron is 1 2 K . Then K is _____.
61. If the angle between two adjacent faces is 1 1 cos K
, then K = _______.
Q. (62-63)
Let OABC be a regular tetrahedron. p,q,r and s be the total surface area, slant height, altitude and volume of the regular tetrahedron of edge length 3 units respectively, then
62. The value of 2 2 22 ps rq −= _____.
63. If maximum value of (log 3 p )cosθ + (log 3 (4 pqrs ))sinθ is k , then [ k ]= _____. Where [.] denotes greatest integer function.
Q. (64-65)
Let r be a position vector of a variable point in Cartesian plane such that OXY plane such that () .10840 rjir−−= and {}{} 22 12max23;min23. prijprijA =+−=+− {}{} 22 12max23;min23. prijprijA =+−=+− tangent line is drawn to the curve 2 8 y x = at A point A with abscissa 2. The drawn line cuts the x-axis at a point B
64. p1 + p2 is equal to ______.
65. . ABOB is equal to ______.
Q. (66-67)
ABCD is a parallelogram L is Point on BC which divides BC in the ratio 1 : 2 AL intersects BD at P. M is point on DC which divides DC in the ratio 1 : 2 and AM intersects BD at Q
66. Point Q divides DB in the ratio m : n where m,n are coprime then n–m= ______.
67. Point P divides AL in the ratio m : n where m,n are coprime then m+n = _____.
Q. (68-70)
Vertices of a parallelogram taken in order are A(2, –1, 4); B(1, 0, –1); C(1, 2, 3) and D.
68. The distance between the parallel lines AB and CD is _____.
(1) 6 (2) 36 5 (3) 22 (4) 3
69. Distance of the point P(8, 2, –12) from the plane of the parallelogram is _____.
(1) 46 9 (2) 326 9 (3) 166 9 (4) 86 9
70. The orhtogonal projections of the parallelogram on the three coordinate planes xy,yz,zx, respectively, are ______.
(1) 14, 4, 2 (2) 2, 4, 14 (3) 4, 2, 14 (4) 2, 14, 4
Q. (71-73)
Vectors ,and xyz each of magnitude 2 make an angle of 60° with each other
71. Vector x is (1) ()() 1 2 abcab −×++
(2) ()() 1 2 abcab +×+−
(3) ()() 1 2 abcab −+×++
(4) ()() 1 2 abcab +×−+
72. Vector y is (1) () 1 2 acbba +×−−
(2) () 1 2 accba −×++
(3) () 1 2 abcba +×++
(4) () 1 2 ababa −×+−
73. Vector z is (1) () 1 2 accba −×−+
(2) () 1 2 abcba +×+−
(3) () 1 2 cabba ×−++
(4) () 1 2 abcba +×++
Matrix Matching Questions
74. Match the items of List-I with the items of List-II..
List - I
(A) If ,and abc are three mutually perpendicular vectors where 2, ab== 1 c = , then abbcca ×××
(B) If and ab are two unit vectors inclined at 3 π , then () 16 ababb +×
List - II
is (p) –12
is (q) 0
(C) If and bc are orthogonal unit vectors and bca ×=
then abcabbc ++++
is (r) 16
(D) If xyaxyb
==
0 abc
, each vector being a non-zero vector, then xyc
is (s) 1 (t) 4
Choose the correct answer from the options given below
(A) (B) (C) (D)
(1) p r s q
(2) r p p q
75. It O is circumcenter and H is orthocenter, G is the centroid of ∆ABC, then match the following: List - I List - II (A) OAOBOC ++
(p) 1 2 HO
(B) HAHBHC ++
(C) AHHBHC ++
(D) OG
(q) 2 HO
(r) 2 AO
(s) 1 3 OH
(t) OH
Choose the correct answer from the options given below
(A) (B) (C) (D)
(1) p t r q
(2) t q s r
(3) t q r s
(4) q s r t
76. If ,, abc are non-coplanar vectors and abc = k, then match the following:
List - I List - II
(A) 234abc (p) 4k
(B) abbcca ××× (q) k2
(C) abbcca +++ (r) 2k
(D) abbcca (s) 24k (t) 0
Choose the correct answer from the options given below
(A) (B) (C) (D)
(1) s q p t
(2) s p r t
(3) s p q t
(4) s q r t
77. Match items of List-I with the items of List-II.
List - I
(A) Volume of parallelepiped determined by vectors
, ab and c is 2.
Then the volume of the parallelepiped determined by vectors
()()2,3abbc ×× and
() ca × is
(B) Volume of parallelepiped determined by vectors
, ab and c is 5.
Then the volume of the parallelepiped determined by vectors
()()3,abbc ++ and
() 2 ca + is
(C) Area of a triangle with adjacent sides determined by vectors
a and b is 20. Then the area of the triangle with adjacent sides determined vectors
() 23ab + and () ab is
(D) Area of a triangle with adjacent sides determined by vectors
a and b is 15. Then the area of the parallelogram with adjacent sides determined vectors
() ab + and a is
Choose the correct answer from the options given below
(A) (B) (C) (D)
List - II
(p) 100
(q) 30
(1) s q r p
(2) q r p s
(3) r s p q
(4) p s r q
78. Match items of List-I with the items of List-II
List - II
List - I
(A) Given four points A(2, 1, 0), B(1, 0, 1), C(3, 0, 1) and D(0, 0, 2)Point D lies on a line L orthogonal to the plane determined by points A,B,C. If point of intersection of plane ABC and line L is (x0, y0, z0), then (7x0 + 2y0 + 8z0) is equal to
(B) If volume of parallelopiped formed by vectors , abbc ××
(p) 9
(r) 24
(s) 60
(q) 10
and ca × is 25 square units, then the volume of parallelopiped formed by vectors , abbc ++ and ca + is equal to
(C) A variable plane at a distance of 3 2 units
from the origin cuts the coordinate axes at P,Q and R. If the centroid G(u,v,w) of triangle PQR satisfies the relation u–2 + v–2 + w–2 = λ, then λ is equal to
(r) 11
(D) Let 2 ˆ ˆ ˆ ijkα=−++ and ˆ ˆ 2 ˆ ijkβ=−−− be two vectors. The area of parallelogram having diagonals 3α and 2β is equal to (s) 12
BRAIN TEASERS
1. Suppose x,y,z > 0 and let o ˆ ˆ cssin 66 axyiyj ππ
and o ˆ ˆ cssin 33 byzizj ππ
then least value of ab + is (1) 22 xz + (2) 22 yz + (3) 22 xy + (4) 222 xyz ++
2. Let , , abc be three non–coplanar vectors, Let S i ( i = 1, 2, 3, 4, 5, 6) de notes the six scalar triple products formed by all possible permultations of , , abc . If i,j,k,l are randomly chosen distinct numbers from 1 to 6 and if , ikik jljl SSSS xySSSS =+=− then (1) 1 (2) 4 (3) 8 (4) 2
3. If the four faces of a tetrahedron are represented by the equations,
Choose the correct answer from the options given below
(A) (B) (C) (D)
(1) r q p s
(2) r q s p
(3) q r s p
(4) q r p s
Numerical Value Questions
4. Consider the set of eight vectors {{}} ˆ :,,1 ˆ , ˆ 1. Vaibjckabc=++∈− Three non-coplanar vectors can be chosen from V in 2n ways. Then, n is _____.
5. Let , and abc be three non–coplanar unit vectors such that the angle between every pair of them is . 3 π If abbcpaqbrc ×+×=++
where p,q and r are scalars, then the value of 222 2 2 pqr q ++ is ______.
6. If O be an interior point of ∆ABC such that 230OAOBOC++=
and () , ˆ ˆ
rijkβγαλ ++=
then volume of tetrahedron (in cubic units) (where β , γ , α, λ are positive numbers) is (1) () 3 3 1 6 λ αβγ (2) 3λ αβγ
then the ratio of the area of ∆ ABC to area of ∆ AOC is ______, where O is origin.
7. If , xy are two non–zero and non-collinear vectors satisfying [(a – 2)α2 + (b – 3)α + c]
x + [(a – 2)β2 + (b – 3)β + c] y + [(a – 2)
y2 + (b – 3) γ + c] () xy × = 0, where α, β, γ are three distinct real numbers, then find the value of (a2 + b2 + c2 – 4) = ______.
8. Given a tetrahedron DABC with AB = 12, CD = 6. If the shortest distance between the skew lines AB and CD is 8 and the angl e between them is 6 π . Then the 1 6 of the volume of the tetrahedron is _____.
FLASHBACK (Previous JEE Main Questions)
JEE Main
1. Let () 2, 3 ˆˆˆˆ aijkbijk =++=−+ . Let c be the vector such that and 3. acbac ×=⋅=
Then () () acbbc ⋅×−−
is equa l to (27th Jan 2024 Shift 1) (1) 32 (2) 24 (3) 20 (4) 36
2. Let the image of the point (1, 0, 7) in the line 1 2 123 y xz == be the point ( α , β, γ). Then which one of the following points lies on the line passing through (α, β, γ) and making angles 2 3 π and 3 4 π with y-axis and z-axis respectively and an ac ute angle with x-axis? (27th Jan 2024 Shift 2)
(1) () 1,2,12 −+ (2) () 1,2,12 (3) () 3,4,322 (4) () 3,4,322 −+
3. The position vectors of the vertices A,B and C of a triangle are 233, ˆˆˆˆˆˆ 223 ijkijk −+++ and 3 ˆ ˆ ˆ ijk−++ , respectively. Let I denotes the length of the angle bisector AD of ∠BAC where D is on the line segment BC, then 2l2 equals: (27th Jan 2024 Shift 2) (1) 49 (2) 42 (3) 50 (4) 45
4. Let the position vectors of the vertices A,B and C of a triangle be 22,2 ˆˆˆˆˆˆ 2 ijkijk ++++ and 2 ˆ 2 ˆ ˆ ijk ++ respectively. Let l1, l2 and l3 be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides AB,BC and CA, respectively, then l1 2 + l2 2 + l3 2 equals (27th Jan 2024 Shift 2)
(1) 1 5 (2) 1
5. Let , ab and c be three non-zero vectors such that b and c are non-collin ear if 5 ab + is collinear with ,6 cbc + is collinear with a and 0 abc αβ ++= , then α+β is equal to (29th Jan 2024 Shift 1) (1) 35 (2) 30 (3) –30 (4) –25
6. Let O be the origin and the position vector of A and B be ˆ 2 ˆ ˆ 2 ijk ++ and ˆ 24 ˆ 4 ˆ ijk ++ respectively. If the internal bisector of ∠AOB meets the line AB at C, then the length of OC is (29th Jan 2024 Shift 1) (1) 2 31 3 (2) 2 34 3 (3) 3 34 4 (4) 3 31 2
7. Let ,124 OAaOBab ==+
and , OCb =
where O is the origin. If S is the parallelogram with a djac ent sides OA and OC , then area of the quadrilateral area of OABC S is equal to (29th Jan 2024 Shift 2) (1) 6 (2) 10 (3) 7 (4) 8
8. Let a unit vector
ˆ ˆ uxiyjzk =++ make angles , 23 ππ and 2 3 π with the vectors 1111 , 2222
ikjk ++ and 1 2 ˆ 2
1 ij + , respectively. If 111 ˆ 2 ˆ 22 ˆ vijk =++ then 2 | ˆ | uv is equal to (29th Jan 2024 Shift 2) (1) 11 2 (2) 5 2 (3) 9 (4) 7
9. Let 23123 an ˆ d
i aaiajakbbibjbk =++=++
(1) 1 410 2 (2) 1 474 2 (3) 1 586 2 (4) 1 306 2
11. Let ,, ˆ ˆ ˆ aijkR αβαβ =++∈ Let a vector b be such that the angle between a and b is 4 π and 2 ||6, b = If 32, ab⋅= then the value of ()222 || ab αβ+× is equal to (30th Jan 2024 Shift 2)
(1) 90 (2) 75 (3) 95 (4) 85
12. Let and ab be two vectors such that 1 and 2 bba=×= . Then () ||2 bab ×− is equal to (30th Jan 2024 Shift 2) (1) 3 (2) 5 (3) 1 (4) 4
13. Let
()() 1 :2 ˆˆˆˆˆˆ 2,, LrijkijkR =−++−+∈ λλ
()() 2 ˆˆˆˆ :, ˆ 3 LrjkijpkR µµ =−+++∈ and
be two vectors such that 1; 2 and 4 aabb=⋅==
1; 2 and 4 aabb
. If () 23 cabb =×−
then the angle between and bc is equal to: (30th Jan 2024 Shift 1) (1) 1 2 cos 3
(3) 1 3 cos 2
10. Let A(2, 3, 5) and C(–3, 4, –2) be opposite vertices of a parallelogram ABCD , if the diagonal ( BD ) 2 ˆ ˆ 3 ˆ BDijk =++
then the area of the parallelogram is equal to (30th Jan 2024 Shift 1)
()
3 ˆ : ˆ ˆ LrimjnkR δδ =++∈ be three lines such that L1 is perpendicular to L2 and L 3 is perpendicular to both L1 and L2. Then the point which lies on L 3 is (30th Jan 2024 Shift 2)
(1) (–1, 7, 4) (2) (–1, –7, 4) (3) (1, 7, –4) (4) (1, –7, 4)
14. Let ˆ 32, ˆˆˆˆ ˆ 47 aijkbijk =+−=++ and 34 ˆ ˆ ˆ cijk =−+ be three vectors. If a vectors p satisfies pbcb ×=× and ·0pa = , then
()
ˆ ˆ · ˆ pijk is equal to (31st Jan 2024 Shift 1)
(1) 24 (2) 36 (3) 28 (4) 32
15. The distance of the point Q(0, 2,–2) form the line passing through the point P (5, –4, 3) and perpendicular to the lines ()() 2 ˆˆ ˆ ˆˆ 3235, rikijkλλ =−++++∈ and ()() 3 ˆˆˆˆˆˆ 2 2,rijkijkµµ =−++−++∈ is (31st Jan 2024 Shift 1) (1) 86 (2) 20 (3) 54 (4) 74
16. Let ˆ 53,4 ˆ 2 ˆˆˆˆ aijkbijk =−+−=+−
and
() ()ˆˆˆ cabiii =×××× . Then
ˆ ˆ ˆ cijk −++ is equal to (1st Feb 2024 Shift 1) (1) –12 (2) –10 (3) –13 (4) –15
17. Consider a ∆ ABC where A (1, 2, 3), B (–2, 8, 0) and C(3, 6, 7). If the angle bisector of ∆BAC meets the line BC at D, then the length of the projection of the vector AD on the vector AC
is (1st Feb 2024 Shift 2) (1) 37 238 (2) 38 2 (3) 39 238 (4) 19
18. The least positive integral value of α , for which the angle between the vectors ˆ ˆ 22 ijkα−+ and ˆ ˆ 22 ijkαα+− is acute, is _. (27th Jan 2024 Shift 1)
19. Let O be the origin, and M and N be the points on the lines 4 55 413 g mz == and 2 8 11 1259 g mz + ++ == respectively such that MN is the shortest dista nc e between the given lines. Then OMON
is equal to ______. (29th Jan 2024 Shift 2)
20. Let Q and R be the feet of perpendiculars from the point P ( a,a,a ) on the lines x = y , z =1 and x = – y , z = –1 respectively. If ∠ QPR is a right angle, then 12a2 is equal to ______. (31st Jan 2024 Shift 1)
21. Let and ab be two vectors such that 1,4ab== and ·2.ab = If
()23 cabb =×− and the angle between and bc is α, then 192sin2α is equal to _____. (31st Jan 2024 Shift 1)
22. Let ˆ 32 , ˆˆˆˆ 2 ˆ 3 aijkbijk =++=−+ and c be a vector such that
()() ˆ 226 ˆ 4 abcabjk +×=×+− and () 3 ˆ abic−+⋅=− . Then ||2 c is equal to ______.
(31st Jan 2024 Shift 2)
23. Let the line of the shortest distance between the lines
ˆˆˆˆˆˆ :23 Lrijkijk λ =+++−+
()() 1
()() 2 ˆˆˆˆˆˆ :456 Lrijkijk µ =++++− intersect L1 and L2 at P and Q respectively. If (α, β, γ) is the midpoint of the line segment PQ, then 2(α + β + γ) is equal to ______.
(1st Feb 2024 Shift 1)
24. Let 8 ˆˆˆˆˆˆ ,2 aijkbijk =++=−−+ and 23 ˆ 4 ˆ ˆ cicjck =++ be three vectors such that baca ×=× . If the angle between the vector c and the vector ˆ ˆ 34ijk ++ is q then the greatest integer less than or equal to tan2 q is _____.
(1st Feb 2024 Shift 2)
25. Let 6912, ˆ 1 ˆˆ ˆ 2 ˆ 1 ˆ aijkbijk α =++=+− and c be vectors such that . acab ×=× If () .12,.25, ˆ ˆ ˆ accijk=−−+= then () ˆ ˆ ˆ cijk ++ is equal to_____. (8th Apr 2023 Shift 1)
26. Let 23 ˆ ˆ ˆ aijk =++ and ˆ ˆ ˆ bijk =+− . If c is a vector such that ·11ac = , () ·27bac×=
and ·3 bcb =− , then ||2 ac × is equal to _. (11th Apr 2023 Shift 2)
27. Let ˆ ˆ ˆ 3 aijk =+− and 33 ˆ 2 ˆ ˆ cijk =−+
. If b
is a vector such that abc =×
and 2 50, b =
then 2 72 bc−+ is equal to _______. (13th Apr 2023 Shift 1)
28. Let ˆ 2,35,7, ˆ ˆˆ ˆ aijkbijkac λλ =++=−−⋅=
2430,. bcacbc ⋅+=×=×
Then ab
is equal to ________. (24th Jan 2023 Shift 2)
29. Let , and abc be three non–zero coplanar vectors. Let the position vectors of four points A,B,C and D be ,34,23 abcabcabc λ −+−+−+−
and 246 abc −+
respectively. If , and ABACAD
are coplanar, then λ is equal to _____. (29th Jan 2023 Shift 1)
30. If λ1 < λ2 are two values of λ such that the angle between the planes () 1 :. 35 ˆ ˆ ˆ 7 Prijk−+= and () 2 ˆ :. 3 ˆ ˆ 9 Prijkλ+−= is 1 26 sin, 5
then the square of the length of perpendicular from the point (38λ1, 10λ2, 2) to the plane P1 is ______. (30th Jan 2023 Shift 1)
31. Let θ be the angle between the planes () 1 :29 ˆ . Prijk++= and
() 2 : 15 ˆ .2 Prijk−+= .
Let L be the line that meets P2 at the point (4, –2, 5) and makes an angle θ with the normal of P2. If α is the angle between L and P2, then (tan2θ)(cot2α) is equal to ______.
(31st Jan 2023 Shift 1)
32. Let and ab be two vectors such that 14,6 and 48 abab ==×= . Then
()2 ab ⋅ is equal to ______.
(31st Jan 2023 Shift 1)
33. Let ,, abc be three vectors such that 31,42abc=== and
()()23abca ×=× . If the angle between and bc is 2 3 π then 2 ac ab × ⋅ is equal to ______.
(31st Jan 2023 Shift 2)
34. Let 23,2 and ˆˆˆˆˆˆ vijkwijku αα =+−=+− and be a vector such that 0 u α => . If the minimum value of the scalar triple product uvw
is 3401 α , and 2 | ˆ | m ui n ⋅= . where m and n are coprime natural numbers, then m + n is equal to ______.
(1st Feb 2023 Shift 1)
35. A(2, 6, 2), B(–4, 0, λ), C(2, 3, –1) and D(4, 5, 0), |λ| ≤ 5 are the vertices of a quadrilateral ABCD . If its area is 18 square units, then 5 – 6λ is equal to _____.
(1st Feb 2023 Shift 2)
JEE Advanced
36. Let the position vectors of points P,Q,R and S be 25,36, ˆˆ 3 ˆˆˆˆ aijkbijk =+−=++
5 ˆ ˆ ˆ 1716 7 5 cijk =++ and ˆ
ˆ 2 dijk =++ , respectively. Then which of the following statements is true? (2023 P2)
(1) The points P,Q,R and S are NOT coplanar
(2) 2 3 bd + is the position vector of a point which divides PR internally in the ratio 5 : 4
(3) 2 3 bd + is the position vector of a point which divides PR externally in the ratio 5 : 4
(4) The square of magnitude of the vector bd × is 95
37. Let ˆ
ˆ , and ijk be the unit vectors along the three positive coordinate axes. Let
38. Let O be the origin and 22, ˆˆˆˆ 2 ˆˆ 2 OAijkOBijk =++=−+
and () 1 2 OCOBOA =−λ
for some λ > 0. I f 9 2 OBOC×=
, then which of the following statements is(are) True? (2021 P2)
(1) Projection of OC
on OA
is 3 2
(2) Area of the triangle OAB is 9 2
(3) Area of the triangle ABC is 9 2
(4) The acute angle between the diagonals of the parallelogram with adjacent sides and is 3 OAOC π
be three vectors such that 23 0,0bbab>⋅=
39. Let a and b be positive real numbers. Suppose ˆ ˆ PQaibj =+ and ˆ ˆ PSaibj =− are adjacent sides of a parallelogram PQRS. Let and uv be the projection vectors of ˆ ˆ wij =+ along PQ and , PS respectively. If uvw += and if the area of parallel ogram PQRS is 8, then which of the following stat ements is/ are TRUE? (2020 P2)
(1) a + b = 4
Then, which of the following is/are TRUE? (2022 P2)
(2) a – b = 2
(3) The length of the diagonal PR of the parallelogram PQRS is 4
(4) w is an angle bisector of the vectors and PQPS
40. Let L1 and L2 denote the lines () ˆ ˆ 22, ˆˆ riijkR λλ =+−++∈ and () 2 ˆ ˆ ˆ 2, rijkR µµ =−+∈ respectively. If L 3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of the following options describe(s) L3? (2019 P1)
(1) ()() ˆ 1 22 ˆ 2, 3 ˆ ˆ ˆ riktijktR =+++−∈
(2) ()() 2 422, ˆ 9 ˆˆ ˆ ˆˆ rijktijktR =++++−∈
(3) ()() 2 ˆˆˆˆ 2 222, 9 ˆˆ rijktijktR =−+++−∈
(4) () ˆ ˆ 22, ˆ rtijktR =+−∈
41. Three lines 1 , ˆ :, LriR λλ =∈ 2 ˆ :, ˆ LrkjR µµ =+∈ and 3 ˆ :, ˆ ˆ LrijvkvR =++∈ are given. For which point (s) Q on L2 can we find a point P on L1 and a point R on L 3 so that P,Q and R are collinear? (2019 P2) (1) ˆ 2 ˆ 1 kj (2) ˆ k (3) ˆ 2 ˆ 1 kj + (4) ˆ ˆ kj +
42. Let ˆ ˆ ˆ 2 aijk =+− and 2 ˆ ˆ ˆ bijk =++ be two vectors. Consider a vector cab αβ=+ , where α, β ∈ R. If the projection of c on the vector () ab + is 32 , then the minimum value of (() ) . cabc −× equals to _____. (2019 P2)
43. In a triangle PQR, let , aQRbRP ==
and cPQ = . If 3,4ab== and () () , acba cabab ⋅− = +
, then the value of ||2 ab ×
is ________. (2020 P1)
44. Let , and uvw be vectors in threedimensional space, where and uv are unit vectors which are not perpendicular to each other and 1,1,4uwvwww ⋅=⋅=⋅=
45. Let P be the plane 32316 xyz++= and let { 222 ˆ : ˆ ˆ 1 Sijkαβγαβγ =++++= and the distance of (α, β, γ) from the plane P is 7 }. 2 Let , and uvw be three distinct vectors in S such that uvvwwu −=−=− . Let V be the volume of the parallelepiped determined by vectors , and uvw . Then the value of 80 3 V is ______. (2023 P1)
46. Let I 1 and I 2 be the lines () 1 ˆ ˆ ˆ rijk λ =++ and ()() 2 ˆˆ ˆ ˆ rjkik µ =−++ respectively, Let X be the set of all the planes H that contain the line I 1. For a plane H , let d ( H ) denote the smallest possible distance between the points of I2 and H. Let H 0 be a plane in X for which d(H0) is the maximum value of d(H) as H varies over all planes in X. Match each entry in List-I to the correct entries in List-II (2023 P1)
List - I
(A) The value of d(H0) is (p) 3
(B) The distance of the point (0, 1, 2) from H 0 is (q) 1 3
(C) The distance of origin from H 0 is (r) 0
(D) The distance of origin from the point of intersection of planes y = z, x = 1 and H 0 is (s) 2 (t) 1 2
(A) (B) (C) (D)
(1) q s t p
(2) t s r p
(3) q p r q
List - II
Choose the correct answer from the options given below
. If the volume of the parallelepiped, whose adjacent sides are represented by the vectors , and , is 2 uvw , then the value of 35uv + is _____. (2021 P1)
(4) t p s q
CHAPTER TEST–JEE MAIN
Section – A
Single Option Correct MCQs
1. Let ,and abc be three non-zero vectors such that .0bc = and () 2 abcbc ××=
If d be a vector such that ()()..,then. bdababcd =××
, then is equal to (1) 3 4 (2) 1 2 (3) 1 4 (4) 1 4
6. Let ˆ 434 ˆ ˆ 5 ˆˆ ,3 aijbijk =+=−+ . If c is a vector such that () 250 cab⋅×+= , () 4 ˆ ˆ ˆ cijk⋅++= , and Projection of on ca is 1, then the projection of on cb equals (1) 1 2 (2) 3 2 (3) 5 2 (4) 1 5
7. If ,, abc are three non-zero vectors and ˆ n is a unit vector perpendicular to c such that () 0 ˆ ,abnαα =−≠ and .12bc = , then
2. Let ,1 ˆ ˆ ˆ aijkab=−−+⋅= and ˆ ˆ abij ×=−
then 6 ab is equal to (1) () ˆ 3 ˆ ˆ ijk ++ (2) () ˆ 3 ˆ ˆ ijk +− (3) () ˆ 3 ˆ ˆ ijk −+ (4) () ˆ 3 ˆ ˆ ijk
3. If the four points, whose position vectors are 342,2,23
ijkijkijk −++−−−+ and 52 ˆ ˆ ˆ 4 ijk α −+ are coplanar, then α is equal to (1) 73 17 (2) 73 17 (3) 107 17 (4) 107 17
4. If the vectors 4,24 ˆˆˆˆ ˆ 2 ˆ aijkbijk λµ =++=−+−
and ˆ 3 ˆ 2 ˆ cijk =++ are coplanar and the projection of a on the vector b is 54 units, then the sum of all possible values of λ + μ is equal to (1) 18 (2) 0 (3) 24 (4) 6
5. If ˆ 2,,734, ˆˆˆ ˆ ˆˆˆ aikbijkcijk =+=++=−+
0 rbbc×+×=
and 0. ra⋅=
Then rc
is equal to (1) 36 (2) 34 (3) 30 (4) 32
()cab ×× is equal to: (1) 9 (2) 12 (3) 6 (4) 15
8. Let , ˆ ˆ ,232 ˆ ˆˆ ˆ aijkbijkλλλ ∈=+−=−+ . If ()() ()() 84024 ˆ ˆ ˆ abababijk +×××−=−−
, then ()() ||2 ababλ+×− is equal to (1) 132 (2) 136 (3) 144 (4) 140
9. Let a and b be two vectors, Let 1,4ab== and 2. ab⋅= If ()23 cabb =×− , then the value of . bc is (1) –24 (2) –84 (3) –60 (4) –48
10. Let 2 ˆ aijk =++ and and bc be two nonzero vectors such that abcabc ++=+− and 0 bc⋅=
Consider the following two statements:
A) aca λ +≥ for all λ ∈ R.
B) and ac are always parallel. Then (1) both (A) and (B) are correct (2) only (B) is correct (3) only (A) is correct (4) neither (A) nor (B) is correct
11. Let 23,2 ˆˆˆˆˆˆ aijkbijk =++=−+ and 33 ˆ 5 ˆ ˆ cijk =−+ be three vectors. If r is a vector such that, rbcb ×=× and 0, ra⋅= then 25||2 r is equal to (1) 560 (2) 339 (3) 449 (4) 336
12. Let ˆ ˆ 53 ˆ aijk =−− and 35 ˆ ˆ ˆ bijk =++ be two vectors. Then which one of the following statements is TRUE?
(1) Projection of on ab is 17 35 and the direction of the projection vector is opposite to the direction of b
(2) Projection of on ab is 17 35 and the direction of the projection vector is same as b
(3) Projection of on ab is 17 35 and the direction of the projection vector is opposite to the direction of b
(4) Projection of on ab is 17 35 and the direction of the projection vector is same as of b
13. Let ˆˆ ˆ ˆˆ 275, aijkbik =−+=+ and 23 ˆ ˆ ˆ cijk =+− be three given vectors. If r is a vector such that raca ×=× and 0, rb⋅=
then r is equal to:
(1) 11 7 (2) 914 7 (3) 11 2 7 (4) 11 2 5
14. Let G 1 , G 2 , G 3 be the centroids of the triangular faces OBC,OCA,OAB of a tetrahedron OABC. If V1 denotes the volume of the tetrahedron OABC and V 2 denotes the volume of the paralellopiped with OG1, OG2, OG3 as three concurrent edges, then (1) 4V1 = 9V2 (2) 9V1 = 4V2
(3) 3V1 = 2V2 (4) 3V1 = 2V2
15. , , abc are three unit vectors. b is not parallel to c and () 1 2 abcb ××= , then ()() ,; , abac==
(1) 90°, 60° (2) 45°, 45° (3) 60°, 30° (4) 90°, 45°
16. If , ab are non–zero and non–collinear vectors then abiiabjjabkk ++=
(1) ab × (2) ba ×
(3) 0 (4) . ab
17. If , , abc are non–coplanar, nonzero vectors and r is any vector in space, then abrcbcracarb ++=
(1) 3 abcr (2) abcr (3) abc (4) 0
18. A vector () ˆ 2, ˆ ˆ , aijkR αβαβ =++∈ lies in the plane of the vectors ˆ ˆ bij =+ and ˆ 4 ˆ ˆ cijk =−+ . If a bisects the angle between b and c , then
(1) .10 ˆ ai += (2) .30 ˆ ai +=
(3) .40 ˆ ak += (4) .20 ˆ ak +=
19. Let the volume of a parallelepiped whose coterminous edges are given by ˆ , ˆˆˆˆ 3 ˆ uijkvijk λ =++=++ , and ˆ ˆ ˆ 2 wijk =++ , be 1 cubic unit. If θ be the angle between the edges and uw , then cosθ can be: (1) 7 63 (2) 7 6 6 (3) 5 7 (4) 5 33
20. Let 2 ˆ ˆ ˆ aijk =−+ and ˆ ˆ ˆ bijk =−+ be two vectors. If c is a vector such that bcba ×=×
and ·0ca = , then cb is equal to (1) 3 2 (2) 1 2 (3) 1 2 (4) –1
Section – B
Numerical Value Questions
21. If the vectors, () () ˆ 1, ˆ ˆ 1 ˆ ˆ paiajakqaiajak =+++=+++
and ()() ˆ ˆ 1 ˆ raijakaR =+++∈ are coplanar and 22 3(·)||0 pqrq λ −×= , then the value of λ is _____.
CHAPTER TEST – JEE ADVANCED
2018 P1 Model Section – A [Multiple Option Correct MCQs]
1. Let PQR be a triangle. Let ,, aQRbRPcPQ === . If 12,43ab== and .24bc = , then which of the following are true ______.
(1) 2 12 2 c a −= (2) 2 30 2 c a += (3) .2ab =− (4) .72ab =−
22. Points A(2, 4, 0), B(3, 1, 8), C(3, 1, –3), D(7, –3, 4) are four points. The projection of the line segment AB on line CD is ______.
23. Let , and abc be three vectors such that 3,5,·10abbc=== and the angle between and bc is 3 π . If a is perpendicular to the vector bc × , then ()abc ×× is equal to _____.
24. Let , , abc be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle θ, with the vector abc ++ . Then, 36cos22q is equal to ______.
25. If the shortest distance between the lines () 1 ˆ 222 ˆ ˆ 2 ˆ ˆ , ˆ rijkijk αλ =+++−+ λ ∈ R ,
> 0 and () 2 4322, ˆˆ ˆ ˆˆ rikijk µ =−−+−− μ ∈ R is 9, then α is equal to ______.
2. The vector ˆ ˆ ˆ 3 ixjk ++ is rotated through an angle q and is double in magnitude.It now becomes ()442 ˆ ˆ 2 ˆ ixjk +−+ the values of x are (1) 1 (2) 2 3 (3) 2 (4) 4 3
3. Let OA,OB,OC be coterminous edges of a cuboid. If l,m,n be the shortest distances between the sides OA,OB,OC and their respective skew body diagonals to them respectively, them the value of
is less than
12: Vector Algebra
(1) 2 (2) 3 (3) 4 (4) 5
4. Let , ˆˆˆˆˆˆ 334. ABijkACijk =+−=++ and ˆ ˆ 2 ADik =− are three co–terminus edges of a tetrahedron ABCD. If position vector of the centroid of tetrahedron is () ˆ 3 ˆ 2 ˆ ijk ++ then
(1) Position vector of point A is () 2 ˆ 3 ˆ ik +
(2) Length of perpendicular from D to the plane ABC is 128 111
(3) Acute angle between the face ABC and ACD is 1 146 cos 27972
(4) Volume of tetrahedron is 8 3
5. Let ()() 1 ˆ :23725 ˆ ˆˆˆˆ ,, LrijkipjkR λλ =−++++∈ ():23725 ˆ ˆˆˆˆ ,, LrijkipjkR λλ =−++++∈
2 ˆˆ ˆ ˆˆ :43, LrikipjpkR µµ =−+−+∈
And if they are perpendicular to each other for all values of λ and m , then which of the following may be true:
(1) p =6
(2) p2 + 7p + 6 = 0
(3) The equation of the plane containing L1 and L2 is 4x – 17y – 5z – 24 = 0 for one of the possible values of ‘’ p’’ (4) L1, L2 are non–coplanar lines for one of the possible values of ‘’ p’’
6. Let ,,, OAaOBbOCcODd ==== and tan2tan2tan, abcd αβγ++= also the points A,B,C,D are coplanar then which of the following options is/are correct?
(1) The maximum value of
222 1 tantantan 4 αβγ ++=
(2) The minimum value of
222 1 tantantan 9 αβγ ++=
(3) At the time of
222 1 tantantan 9 αβγ ++= , then 5 tantantan 9 αβγ++=
(4) At the time of
222 1 tantantan 4 αβγ ++= then 5 tantantan 4 αβγ++=
Section – B
[Numerical Value Questions]
7. Let ,and abc be three non-coplanar vectors, and d be a non-zero vector, which is perpendicular to () . abc ++ Now if ()()() sincos2, dxabybcca =×+×+×
then minimum value of x2 + y2 is equal to 2 , 4 λπ where λ is _______.
8. If 3, ˆˆˆˆˆˆ aijkbijk αββα =++=−−− and 2 ˆ ˆ ˆ cijk =−− such that ·1ab = and ·3,bc =− then () () 1 · 3 abc × is equal to __.
9. Let u be a vector on rectangular coordinate system with sloping 60°. Suppose that ˆ ui is geometric mean of u and 2 ˆ ui where ˆ i is the unit vector along x–axis then u has the value equal to ab where a,b ∈ N, then the value (a + b)3 + (a – b)3 is _____.
10. Let , ˆˆ ˆ 3 ˆ ˆˆ aijkbijk αββα =−+=+− and ˆ ˆ 2 ˆ cijk α =−−+ , where α and β are integers. If 1 ab⋅=− and 10 bc⋅= , then ()abc ×⋅ is equal to ______.
11. Let P,Q,R be the points with position vectors 2 ˆˆˆˆ ˆ 32 ˆ ,34 rijkrijk =−−=++ and 3 ˆ ˆ 2 ˆ 2 rijk =+− relative to an origin O. The distance of P from the plane OQR is ______. (magnitude)
12. If the shortest distance between the lines
ˆ 3 ˆ ˆ ˆ rikiaj λ =−++− and
, ˆˆˆˆ ˆ 2 2 is 3 rjkijk µ =−++−+ then the integ ral value of a is equal to __.
13. A vector of magnitude 3, bisecting the angle between the vectors ˆ ˆ ˆ 2 aijk =+− and 2 ˆ ˆ ˆ bijk =−+ and making an obtuse angle with b is ()
ˆ ˆ ijmk n +− , where l+m+n = _____.
14. Let 6912,112and aijkbijkc α =++=+− 6912,112and aijkbijkc α =++=+− be vectors such that acab ×=× if () .12 and .25accijk=−−+= , then () . cijk +− is ______
Q:(15-16)
Section – C
[Passage-based Questions]
Let ,, ijk be unit vectors along three positive coordinate–axes, 3,,23123 aijkbibjbkccicjck =+−=++=++ .0ab = and 32 1 3 122 2133 013 01 01 ccc ccbc ccbc
then
15. The value of bc = ______.
16. The maximum value of 2 13 2 c = ______.
Q: (17-18)
Consider a line ()ritijk =+++ and a plane (() )() .1rijijk −+−−=
17. The angle between the line and the plane is (1) 1 1 cos 3
(2) 1 1 sin 3
(3) 4 π (4) 2 π
18. The position vector of the point of intersection of the line and the plane is (1) i (2) j (3) k (4) ij +
ANSWER KEY
– II
Level – III
Theory-based Questions
JEE Advanced Level
Brain Teasers
Chapter Test – JEE Main
Chapter Test – JEE Advanced
Chapter Outline
13.1 Direction Cosines and Direction Ratios
13.2 Plane in 3D Geometry
13.3 Straight Line in Space
13.4 Line and Plane
In the previous classes, we explored analytical geometry with a focus on two-dimensional Cartesian methods. Now, as we step into the realm of three-dimensional geometry, we’re incorporating vector algebra to add elegance and simplicity to our studies. This chapter dives into advanced topics, such as direction cosines, direction ratios of lines, and equations for lines and planes in space. We’ll delve into measuring angles between lines and planes, finding the shortest distances between skew lines, and determining distances from points to planes.
13. 1 DIRECTION COSINES AND DIRECTION RATIOS
In this chapter, we shall discuss direction cosines and direction ratios of a line, angle between two lines, projection of a line segment, direction cosines of angle bisectors and the symmetric form of the equation of a straight line in space.
13.1.1 Direction Cosines of a Line in Space
If a directed line OP makes angles ,, αβγ with the positive direction of the axes OX, OY and OZ, respectively, then cos,cos,cos αβγ
3D GEOMETRY CHAPTER 13
are called direction cosines of OP Direction cosines are denoted by ,, lmn
Here, cos,cos,coslmn =α=β=γ
If ,, lmn are direction cosines of a line L , then ,, lmn are also direction cosines of the same line L
The angles ,, αβγ are called direction angles of a line and satisfy the condition 0,,, ≤αβγ≤π
The direction cosines of x-axis are 1,0,0 , the direction cosines of y-axis are 0,1,0 and the direction cosines of z-axis are 0,0,1
Coordinates of a point on directed line: Let direction cosines of line OP
be ,, lmn and let P be a point on the ray OP
at a distance r units from the origin. The coordinates of P are () ,, lrmrnr . If the coordinates of a point P are () ,, xyz and , OPr = then the direction cosines of line OP
are ,, y xz rrr
Relation between the direction cosines of a line:
If ,, lmn are direction cosines of a line L then 222 1 lmn++=
If ,, αβγ are the angles made by a line L with positive direction of axes, then
13: 3D Geometry
222 coscoscos1 α+β+γ= 222 sinsinsin2 α+β+γ= cos2cos2cos21 α+β+γ=−
Direction cosines of a line which makes equal angles with coordinate axes are 111 ,, 333 ±±± .
There are four lines passing through the origin , which makes equal angles with coordinate axes and direction cosines of those lines are 111111111 ,,,,,,,, 333333333 and 111 ,, 333
There are eight rays passing through the origin, which makes equal angles with coordinate axes in space.
If ,, αβγ are the angles made by a line with positive direction of axes, then ,, π−απ−βπ−γ are also the angles made by the same line with axes.
For every line, two sets of ordered triads will be there as direction cosines of that line in space.
The coordinates of two points on a line passing through the origin, which are at 1 unit distance from origin, are direction cosines of that line.
If ,, lmn are direction cosines of a line, then the maximum value of lmn is 1 33
1. If a ray makes angles , 33 ππ with , oxoy
axes, respectively, then find the angle made by the ray with oz
Sol: Given , 33 ππα=β=
We know that 222 coscoscos1. α+β+γ= Hence, 222 2 2 coscoscos1 33 11 cos1 44 1 cos 2 ππ ++γ= ++γ= γ=
It implies that 1 cos 2 γ=±
Therefore, the angle made by the ray with oz is 4 π or 3 4 π
Try yourself:
1. If 111 ,, ccc represents direction cosines of a line, then find the value of c. Ans: 3 ±
13.1.2 Direction Ratios of a Line in Space
Any three numbers that are proportional to the direction cosines of a line are called direction ratios of that line.
Let ,, lmn be the direction cosines of a line L . If the direction cosines of a line are proportional to ,,,abc then ,, abc are called direction numbers or direction ratios of the line.
If ,, lmn are direction cosines of a line, then for any non-zero real number λ , the numbers ,, lmnλλλ are also the direction ratios of the same line.
If ,, abc are direction ratios of a line, then 222 abc ++ need not be equal to 1. Direction cosines of the same line are
222222222 ,, abc abcabcabc ±±± ++++++
For any line, there exists infinite sets ordered triads as direction ratios of that line. The coordinates of any points on the line, which is passing through the origin, are the direction ratios of that line. The direction cosines of a line are also its direction ratios, but not vice versa. The direction ratios of
a line joining the points () 111 ,, Axyz and () 222 ,, Bxyz are 212121,,. xxyyzz
Direction cosines of line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz are 212121 ,,. xxyyzz ABABAB
Collinear Points: Let () 111 ,, Axyz , () 222 ,, Bxyz and () 333 ,, Cxyz be any three points. If the direction ratios of AB and BC are proportional, then the three points ,, ABC are said to be collinear.
2. Find the direction cosines of a line joining the points () 4,1,7 and () 2,3,2
Sol: The direction ratios of the line joining the points () 4,1,7 and () 2,3,2 are 24,31,276,4,5 +−−−=−−
The direction cosines are 645 ,, 361625361625361625 ++++++
Simplifying, 645 ,, 777777
Try yourself:
2. If the direction ratios of a line are () 3,4,0 then find the angle made by the line with z-axis.
Ans: 90°
13.1.3 Angle between Two Lines in Space
Suppose, the direction cosines of two lines are 111 ,, lmn and 222 ,, lmn . If θ is the acute angle between the two given lines, then 121212 cos llmmnnθ=++
■ ()2 1221 sin lmlmθ=∑−
■ ()2 1221 121212 tan lmlm llmmnn ∑− θ= ++ , when 2 θ≠π
The condition that the lines are to be perpendicular is 121212 0. llmmnn++= The
condition that the lines are to be parallel is 111 222 lmn lmn ==
Suppose, the direction ratios of two lines are 111 ,, abc and 222 ,, abc . If θ is the acute angle between the above two lines, then 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++
The condition that the lines are to be perpendicular is 121212 0 aabbcc++= . The condition that the lines are to be parallel is 111 222 abc abc == .
Lagrange’s identity: Let () 111 ,, lmn and () 222 ,, lmn be two ordered triads.
llmmnn lmlmmnmnnlnl ⇒++++ −++ =−+−+−
Points to Remember
1. If a ray makes angles ,,, αβγδ with the four diagonals of a cube, then 2222 4 coscoscoscos 3 α+β+γ+δ=
2. If 111 ,, lmn and 222 ,, lmn are direction cosines of two intersecting lines, then the direction ratios of angular bisectors of the above two lines are 121212 ,, llmmnn ±±± .
3. If 111 ,, lmn and 222 ,, lmn are direction cosines of two lines then the direction cosines of angular bisectors of the above two lines are 121212 ,, 2cos2cos2cos 222 llmmnn +++ θθθ and 121212 ,, 2sin2sin2sin 222 llmmnn θθθ , where θ is an angle between the lines.
4. If 111,,, lmn 222,,, lmn and 333 ,, lmn are the direction cosines of three mutually perpendicular lines, then direction ratios of a line, which makes equal angles with given three lines, are 123123123 ,, lllmmmnnn ++++++ and its direction cosines are 123123123 ,, 333 lllmmmnnn ++++++
5. If 111 ,, lmn and 222 ,, lmn are direction cosines of two lines, then the direction ratios of a line whic h is perpendicular to both the given lines are 122112211221 ,, mnmnnlnllmlm
6. The lines whose direction cosines are related with relations 0 albmcn++= and 0 fmngnlhlm++= are
• perpendicular if 0 fgh abc ++= ,
• parallel if a2f2 + b2g2 + c2h2 – 2abfg – 2bcgh – 2acfh = 0 (or) 0 afbgch±±= .
7. The lines whose direction cosines are related with relations 0 albmcn++= and 222 0 ulvmwn++= are
• perpendicular, if ()()() 222 0 avwbuwcuv+++++= ,
• parallel, if 222 0 abc uvw ++= .
8. If the edges of a rectangular parallelopiped are a,b,c then the angle between its diagonals are 222 1 222 cos abc abc ±±± ++ (In the numerator, all the three signs are not either positive or negative.)
9. Angle between the diagonals of a cube is 1 1 cos 3
10. The angle between the a diagonal of a cube and the diagonal of a face of the cube is 1 2 cos 3
11. If a variable line in two adjacent positions has direction cosines (l, m, n) and (l+ δl, m+δm, n+δn) and δθ is the angle between two positions, then (δl)2 + (δm)2 + ( δn)2 = (δθ)2
12. Any point on the line passing through () 111 ,, xyz and having direction cosines () ,, lmn is in the form of () 111 ,, xlrymrznr +++ , where r is the parameter.
3. Find the angle between the lines whose direction cosines are proportional to 2,1,1 and 4,31,31
Sol: Given: direction ratios of two lines are 2,1,1, 4,31,31
Let θ be the acute angle between the above two lines. Then, 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++ ()()() 24131131 cos 4111631233123 83131 624 1 2 +−+−− θ= ++++−+++ +−−− = =
Therefore, angle between the given two lines is 60.θ=°
Try yourself:
3. Show that the lines PQ and RS are parallel where the points are ()()() 2,3,4,4,7,8,1,2,1PQR and () 1,2,5 S .
13.1.4 Projection of a Line Segment in Space
Let A and B are two points and let l be a directed line. Let L and M be the projections of A and B on the line l and let θ be the angle made by the line AB with the line l
If θ is acute, then LM is called the projection of AB on the line l
If θ is obtuse, then LM is called the projection of AB on the line l.
The projection of the line segment AB on the line l is cos AB θ , where θ acute angle between the line AB and the line l
by the line segment on the axes, then 2222 123 . dddd =++
If d is the length of the line segment and 123 ,,and ddd are the projections of the line segment on coordinate planes, then 2222 123 2. dddd =++
4. Find the projection of the line segment joining the points () 2,3,5 A and () 2,1,3 B on xy-plane.
Sol: The length of the projection of AB on xy-plane is ()() ()() 22 12121 22 2213 161642 dxxyy =−+− =−−+−− =+=
Try yourself:
4. If 3,4,and5 are the projections of a line segment on the coordinate axes, then what is the length of the line segment? Ans: 52
TEST YOURSELF
The projection of the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz on the line , l whose direction cosines are ,,,lmn is ()()() 212121 lxxmyynzz −+−+− .
The length of the projection of the line segment joining two points () 111 ,, Axyz and () 222 ,, Bxyz on
■ x-axis is 121 dxx =−
■ y-axis is 221 dyy =−
■ z-axis is 321 dzz =−
■ xy-plane is ()() 22 12121 dxxyy =−+−
■ yz-plane is ()() 22 22121 dzzyy =−+−
■ zx-plane is ()() 22 32121 dxxzz =−+−
If d is the length of the line segment and 123 ,,and ddd are the projections made
1. If a line makes an angle of 4 π with the positive directions of each of x -axis and y-axis, then the angle that the line makes with the positive direction of the z-axis is (1) 2
2. If the direction cosines of a line L are(ab, b, b) and the angle between L and x-axis is 3 π , then a pair of possible values for a, b are (1) 23 , 38 (2) 82 , 33 (3) 2, 5 (4) 3, 4
3. A line makes the same angle θ with each of the x and z-axes. If the angle β , which it makes with the y-axis, is such that sin2 β =3sin2 θ , then cos2 θ = (1) 2 3
4. A line makes angles α , β , and γ with the coordinate axes. If α + β = 2 π , then (cos
α + cos β + cos γ )2 is equal to (1) 1+sin2 α (2) 1–sin2 α (3) 1+cos2 α (4) 1
5. In a triangle ABC, A = (2, 3, 5), B = (–1, 3, 2), and C = (λ, 5, μ). If the median through A is equally inclined with axes, then ( λ , μ) = (1) (14, 20) (2) (7, 10) (3) 7 ,5 2
(4) (19, 7)
6. If OP is equally inclined with the axes, and if the tip of OP is in positive octant and OP = 6, then P = (1) 222 ,, 333
(2) 111 ,, 333
(3) (6, 6, 6) (4) 666 ,, 333
7. If the line joining the points (k, 2, 3), (1, 1, 2) is parallel to the line joining the points (5, 4, –1) and (3, 2, –3), then k = ______.
8. If the line joining the points (2, 3, 4) and (0, 1, 2) is perpendicular to the line joining the points (x, 0, 4) and (7, –4, 3), then x = _____.
9. If the angle between the lines whose direction cosines are 21 , , 212121 c
and 336 , , 545454
is 2 π , then the value of c is _____.
Answer Key (1) 1 (2) 1 (3) 3 (4) 1 (5) 2 (6) 4 (7) 2 (8) 2 (9) 4
13. 2 PLANE IN 3D GEOMETRY
In this topic, we discuss various forms of the equation of a plane, angle between two planes, angle between a plane and a line, and the planes bisecting the angle between two planes.
13.2.1 Equation of Coordinate Planes
A surface in space is said to be a plane surface, or a plane, if all the points of the straight line joining any two points of the surface lie on the surface.
Coordinate Planes
1. The equation of plane determined by x-axis and y-axis is xy-plane, and its equation is 0. z =
2. The equation of plane determined by y-axis and z-axis is xy-plane, and its equation is 0. x =
3. The equation of a plane determined by z -axis and x -axis is xy -plane, and its equation is 0. y =
5. What is the equation of plane which is parallel to xy-plane?
Sol: zk =
Try yourself:
5. What is the equation of plane which is parallel to yz-plane?
Ans: xk =
13.2.2 General Form of Equation of a Plane
Any first-degree equation in terms of ,, xyz represents the equation of a plane.
1) The general equation of plane is 0. axbyczd+++= Here, ()() ,,0,0,0.abc ≠
The ordered triad ( a , b , c ) are direction ratios of normal to the plane. Here, normal is a perpendicular line to the plane.
i) If a = 0, b ≠ 0, c ≠ 0, then the equation 0 byczd++= represents a plane which is parallel to x-axis and perpendicular
to yz-plane
ii) If b = 0, a ≠0, c ≠0 then the equation ax + cz + d = 0 represents a plane which is parallel to y-axis and perpendicular to xz-plane.
iii) If a ≠ 0, b ≠ 0, c = 0, then the equation ax + by + d = 0 represents a plane which is parallel to z-axis and perpendicular to xy-plane.
2) The equation of any plane perpendicular to x -axis or parallel to yz -plane can be taken as x = k. The equation of any plane perpendicular to y-axis or parallel to xzplane can be taken as y = k. The equation of any plane perpendicular to z -axis or parallel to xy-plane can be taken as z = k.
3) The equation of any plane passing through the point () 111 ,, xyz and having direction ratios of normal to the plane ,, abc is ()()() 111 0 axxbyyczz−+−+−= .
4) The equation of the plane passing through the point () 111 ,, xyz and parallel to the plane 0 axbyczd+++= is ()()() 111 0 axxbyyczz−+−+−=
5) The equation of the plane passing through three points
()()() 111222 ,,,,,,and,,333 AxyzBxyzCxyz is
6) The components of the vector ABAC ×
are the direction ratios of normal to the plane containing the points A, B, C. There exists unique plane passing through three non- collinear points. If three or more points lie on the same plane, then those points are said to be coplanar.
7) If the points
()()() 111222333 ,,,,,,,,, xyzxyzxyz and () 444 ,, xyz are colinear, then 414141 212121 313131 0 xxyyzz xxyyzz xxyyzz −−−=
6. Find the equation of the plane passing through the points () 3,1,0 , () 2,1,1 , and () 1,2,5 .
Sol: Equation of the plane passing through the given points is 111 212121 313131 0 xxyyzz xxyyzz xxyyzz −−−= Hence, 31 1210 435 xyz −+ −−=
Expanding the determinant, will get the equation of plane as 7x + y – 5z = 20
Try yourself:
6. Show that the points ()()()() 0,1,0,2,1,1,1,1,1,and3,3,0 are coplanar.
13.2.3 Normal Form of a Plane
The equation of a plane which is at a distance of p units from the origin and whose normal has direction cosines ,, lmn is lxmynzp ++=
The equation of a plane passing through the origin and whose normal has direction cosines ,, lmn is 0 lxmynz++=
The equation of a plane passing through the point () 111 ,, Pxyz and having normal l ine whose direction ratios are ,, abc is ()()() 111 0 axxbyyczz−+−+−= .
The normal form of the plane ax+by+cz+d = 0 is 2222 , abcd xyz aaaa ++=− ∑∑∑∑
when d <0. The normal form of the plane 0 axbyczd+++= is
2222 , abcd xyz aaaa ++= ∑∑∑∑ when 0 d > .
7. If () ,, abc is foot of the perpendicular of origin on the plane π , then find the equation of the plane π .
Sol: Equation of the required plane is ()()() 0 axabybczc−+−+−=
Hence, the equation of the plane is 222 axbyczabc ++=++ .
Try yourself:
7. Find the distance from the origin to the plane 3412150 xyz+−+=
Ans: 15 13
13.2.4 Intercept Form of a Plane
If the plane cuts the x -axis at A ( a ,0,0), the y-axis at B(0,b,0), and the z-axis at C(0,0,c), then a is called x -intercept, b is called y-intercept, and c is called z-intercept of the plane.
The equation of the plane having intercepts ,, abc , respectively, is 1 y xz abc ++= . The intercept form of the plane 0 axbyczd+++= is 1 y xz ddd abc ++=
The intercepts of the plane 0 axbyczd+++= on the axes are ,, ddd abc .
The equation of the plane whose intercepts are k times the intercepts made by the plane 0 axbyczd+++= on corresponding axes is 0 axbyczkd+++= .
If a plane meets the coordinate axes at ,,,ABC and the centroid of the triangle ABC is () ,, pqr , then the equation of the plane is
3 y xz pqr ++=
The area of the triangle formed by the plane 1 y xz abc ++= with
■ x-axis and y-axis is 1 2 ab square units
■ y-axis and z-axis is 1 2 bc square units.
■ x-axis and z-axis is 1 2 ac square units.
If the plane 1 y xz abc ++= cuts the axes at ,,,ABC then the area of the triangle formed by ,, ABC is ()()() 222 1 . 2 abbcca ++
8. Find the equation of the plane having intercepts 1,2,3 on the axes in the general form.
Sol: Equation of the required plane is 1. 123 y xz ++= General form of the plane is 63260. xyz+−−=
Try yourself:
8. If a plane 1 234 y xz ++= cuts the coordinate axes at ,,,ABC then find the area of the triangle ABC
Ans: 61 square units.
13.2.5 Perpendicular Distance from a Point to the Plane
The perpendicular distance from the origin to the plane 0 axbyczd+++= is 222 d abc ++ .
The perpendicular distance from () 111 ,, Pxyz to the plane 0 axbyczd+++= is 111 222 axbyczd abc +++ ++
Two systems of rectangular coordinate axes have the same origin. If a plane cuts
them at a distance ,, abc and 111 ,, abc , respectively, in order, from the origin, then 222222 111 abcabc ++=++
A variable plane is at a constant distance p from the origin and meets the axes at ,, ABC respectively. The locus of the centroid of the triangle formed by ,,, OABC is 2222 9 xyzp ++=
A variable plane is at a constant distance p from the origin and meets the axes at ,,,ABC respectively. The locus of the centroid of the tetrahedron formed by ,,, OABC is 2222 16 xyzp ++=
9. Find the perpendicular distance from a point () 3,4,1 to the plane 222xyz−+=
Sol: The perpendicular distance from a point () 111 ,, Pxyz to the plane 0 axbyczd+++= is 111 222 axbyczd abc +++ ++
Hence, the distance from the point () 3,4,1 to the plane 222xyz−+= is 3822 3 144 = ++ .
Try yourself:
9. Find the distance from a point () 2,3,4 to the plane passing through three points ()()() 1,3,2,5,0,2,and1,1,4
Ans: 1
13.2.6 Ratio in which the Plane Divides the Line Segment
Two points () 111 ,, Axyz and () 222 ,, Bxyz lie on the same side of the plane
0, axbyczd+++= if 111 axbyczd +++ and 222 axbyczd +++ have the same sign. Then the two points () 111 ,, Axyz and () 222 ,, Bxyz lies on the opposite side of the plane 0 axbyczd+++=
If 111 axbyczd +++ and 222 axbyczd +++ have the opposite sign. Then the two points () 111 ,, Axyz and () 222 ,, Bxyz lies on the same side of the plane
0 axbyczd+++=
The ratio in which the plane 0 axbyczd π≡+++= divides the line segment joining the points () 111 ,, Axyz
and () 222 ,, Bxyz is –π111:π222 or 111 222 axbyczd axbyczd +++ +++
10. If the plane 23520 xyz−+−= divides the line segment joining the points () 1,2,3 and () 2,1, k in the ratio 9:11 , then find the value of k
Sol: The given plane is 23520 xyz π≡−+−= and the given points are () 1,2,3 A and ()2,1,.Bk
The ratio in which the plane divides the line segment joining the points and AB is
–π111:π222.
Hence, 261529 435211 k −+− −= −+− It implies, 2 k =−
Try yourself:
10. If the points () 1,1,3 and () 1,0,3 lie on opposite sides of the plane 30,xyzd+++= then find the range of . d
Ans: 78 d <<
13.2.7 Foot of the Perpendicular and Image of a Point on the Plane
If () ,, Qhkl is foot of the perpendicular of () 111 ,, Pxyz on the plane 0 axbyczd+++= , then () 111 111 222 hxkylzaxbyczd abcabc −+++ === ++
If () ,, Qhkl is image of the point () 111 ,, Pxyz with respect to the plane 0 axbyczd+++= , then () 111 111 222 2 . hxkylzaxbyczd abcabc −+++ === ++
If d is the distance from the origin and (l,m,n) are the direction cosines of the normal to the plane passing through the origin, then the foot of the perpendicular is (dl,dm,dn)
The reflection of the plane 0 axbyczd ′′′′+++= in the plane 0 axbyczd+++= is given by ()
222 2 aabbccaxbyczd abcaxbyczd
11. Find the image of the point () 1,2,1 in the plane 2x–y + 3z + 7 = 0.
Sol: Let (h,k,l) be the image of the point (1, 2, 1) w. r. t the plane 2x–y + 3z + 7 = 0. Then, (()()) ()() 22 2212317 121 213 21 hkl −−++ === +−
121 4 213 hkl ===−
7,6,11hkl=−==−
Therefore, the image of the point () 1,2,1 with respect to the given plane is () 7,6,11 . Try yourself:
11. If () 2,3,1 is the foot of the perpendicular from () 4,2,1 to the plane, then find the equation of the plane.
Ans: 2210 xyz−++=
13.2.8 Angle between Two Planes
Angle between two planes is defined as the angle between their normals.
The angle between two planes
1111 0 axbyczd+++= and 2222 0 axbyczd+++= is the angle between two normal lines having direction ratios 111 ,, abc and 222,,.abc
Let θ be the acute angle between two planes 1111 0 axbyczd+++= and 2222 0 axbyczd+++= . Then, 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++ .
The condition that the two planes are to be perpendicular to each other is 121212 0 aabbcc++= . The condition that the two planes are to be parallel to each other is 111 222 abc abc == . If the angle between two planes is 90° , then those planes are called orthogonal planes.
12. If the planes 2370 xyz−−−= and 42590 xykz−++= are parallel to each other, then find the value of 2 56 k + .
Sol: The condition that the two planes 2370 xyz−−−= and 42590 xykz−++= are to be parallel is 113 425k ==− It gives, 6 5 k = Therefore, 2 3666 5656 255 k +=+=
Try yourself:
12. If the angle between the planes 2470xyz+++= and 3260xyz−++= is () 1 cos k , then find the value of k
Ans: 3 98
13.2.9 Parallel Planes
The equation of any plane parallel to the plane 0 axbyczd+++= can be taken as 0. axbyczk+++=
The equation of plane passing through the point () 111 ,, Pxyz and parallel to the plane 0 axbyczd+++= is ()()() 111 0. axxbyyczz−+−+−=
The distance between two parallel planes
1 0 axbyczd+++= and 2 0 axbyczd+++=
is 21 222 . dd abc ++
The equation of plane which is midway between two parallel planes
1 0 axbyczd+++= and 2 0 axbyczd+++=
is 12 0. 2 dd axbycz + +++=
13. Find the distance between the parallel planes 2230 xyz−++= and 44250 xyz−++=
Sol: The given planes are 2230 xyz−++= and 5 220. 2 xyz−++= The required distance is 12 222 5 3 1 2 6 441 dd abc == ++++
Try yourself:
13. Find the equation of parallel plane lying midway between the parallel planes
23670 xyz−+−= and 23670. xyz−++= Ans: 2360 xyz−+=
13.2.10 Equation of Plane with the Given Conditions
The equation of any plane passing through the line of intersection of two planes 1π and 2π can be taken as 12 0 π+λπ= .
T he equation of any plane passing through () 111 ,, Pxyz and parallel to the lines whose direction ratios are 111 ,, abc and 222 ,, abc is 111 111 222 0 xxyyzz abc abc =
The equation of any plane passing through ()() 111222 ,,,,, PxyzQxyz and parallel to the line whose direction ratios are ,, abc is 111 212121 0 xxyyzz xxyyzz abc −−−=
Equation of plane which is perpendicular bisector of the line segment joining () 111 ,, Axyz and () 222 ,, Bxyz is a plane passing through the midpoint of AB and having the line AB as normal.
14. Find the equation of the plane passing through the point () 2,1,2 and also passing through the line of intersection of two planes 35xyz++= and 23. xyz−+=
Sol: The equation of any plane passing through the line of intersection of two planes 35xyz+−= and 23 xyz−+= can be taken as () 35230.xyzkxyz ++−+−+−= This line is also passing through the point (2,1,–2). () () 232541230 220 1 k k k +−−+−−−= −+−= =−
Hence, the equation of the plane is () 35230 35230 420 xyzxyz xyzxyz xy +−+−−+−= ++−−+−+= −+−=
Therefore, the equation of the required plane is 420.xy−+=
Try yourself:
14. Find the equation of the plane passing through the line of intersection of of two planes 20 xy−= and 30 zy−= , perpendicular to the plane 4538. xyz+−=
Ans: 281790 xyz−+=
13.2.11 Angular Bisector of Two Planes
Equation of two planes bisecting the angles between the planes 1111 0 axbyczd+++= and 2222 0 axbyczd+++= are 11112222 222 222 111 222 axbyczdaxbyczd abcabc ++++++ =± ++++ .
Suppose, 12,0.dd > Then, the equation 11112222 222222 111222 axbyczdaxbyczd abcabc ++++++ = ++++ represents acute angular bisector, if 121212 0 aabbcc++< .
Suppose, 12,0.dd > Then, the equation 11112222 222222 111222 axbyczdaxbyczd abcabc ++++++ = ++++ represents obtuse angular bisector, if 121212 0 aabbcc++> . Both angular bisectors of any two planes are perpendicular to each other.
15. Find the equation of the obtuse angular bisector between the planes 32680 xyz−++= and 2230 xyz−++=
Sol: The given planes are 32680 xyz−++= and 2230. xyz−++=
Here, both 12 and dd are positive and 121212 6212200. aabbcc++=++=>
Hence, the obtuse angular bisector is ()() 3268223 9436414 332687223 5430 xyzxyz xyzxyz xyz −++−++ = ++++ −++=−++ −−−=
Therefore, the equation of the plane is 5x − y − 4z −3 = 0.
Try yourself:
15. Find the equation of the bisector of the angle between the planes 2290xyz++−= and 4312130 xyz−++= , containing the origin.
Ans: 251762780 xyz++−=
TEST YOURSELF
1. The equation of the plane passing through the points (2, 1, 3), (1, 1, 5), (3, 3, 4) is
(1) 4x–2y+2z+11 = 0
(2) 4x–3y+2z–11 = 0
(3) 4x+3y+2z+8 = 0
(4) 2x–3y+5z–16 = 0
2. If the direction ratios of a normal to the planes are 1, 2, 3 and distance of plane from (0, 0, 0) is 5, then equation of plane is (1) x+2y+3z = 514
(2) x–2y+3z = 32
(3) x–2y+3z = 514
(4) x+2y+3z = 311
3. If A = (2, 4, 1), B(–1, 0, 1), C = (–1, 4, 2), then the distance of (1, –2, 1) from the plane ABC is
(1) 2 3 (2) 14 13 (3) 4 3 (4) 1
4. A plane passes through (2, 3, –1) and is perpendicular to the line having direction ratios (3, –4, 7). The perpendicular distance from the origin to this plane is (1) 3 74 (2) 5 74 (3) 6 74 (4) 13 74
5. If the foot of the perpendicular from (0, 0, 0) to a plane is (1, 2, 3), then the equation of the plane is
(1) 2x+y+3z = 14 (2) x+2y+3z = 14 (3) x–2y+3z = 14 (4) x+2y–3z = 14
6. A plane passing through (–1, 2, 3) and whose normal makes equal angles with the coordinate axes is (1) x+y+z+4 = 0 (2) x–y+z+4 = 0 (3) x+y+z–4 = 0 (4) x+y+z = 0
7. If the distance from (1, 2, 4) to the plane 2 x+2y–z+k = 0 is 3, where (k>0), then k = ______.
8. If the distance between the parallel planes 4x–6 y+2 z+k = 0 and 2 x–3 y+z+1 = 0 is 14 , then k = ______, where k > 0.
9. Area of the triangle formed by the plane 2 x–3 y+5 z–30 = 0 with the xy plane (in sq. units) is ______.
Answer Key
(1) 2 (2) 1 (3) 2 (4) 4 (5) 2 (6) 3 (7) 7 (8) 30 (9) 75
13. 3 STRAIGHT LINE IN SPACE
In three-dimensional geometry, a straight line is uniquely determined if either coordinate of one point and its direction are given or two points on the line are given. So, the general equation of line is the combined equation of those two planes.
13.3.1 Equation of Coordinate Axes
The general equation of a line: The intersection of two planes is a line. The equation of the line formed by the intersection of two planes
a1x +b1y + c1z + d1 = 0 and a2x +b2y + c2z + d2 = 0 is the combined equation of both planes as a1x +b1y + c1z
Equations of Axes
1. The equation of x-axis is 0.yz ==
2. The equation of y-axis is 0.xz ==
3. The equation of z-axis is 0.xy ==
4. The equation of the line parallel to x-axis is , yqzr == , where ,. qrR ∈
5. Equation of the line parallel to y -axis is , xpzr == , where ,. prR ∈
6. Equation of the line parallel to z -axis is , xpyq == , where ,. pqR ∈
13.3.2 Symmetric Form of Equation of a Line
1. The equation of a line passing through the point () 111 ,, Pxyz and having direction ratios proportional to ,, abc is 111 xxyyzz abc == . This is called symmetric form of a line. i. Vector form of the line 111 xxyyzz abc == is
() 111 rxiyjzktaibjck =+++++
This vector form of a line is passing through the point whose position vector is 111 xiyjzk ++ and parallel to the vector aibjck ++
13: 3D Geometry
ii. Parametric form of line
111 xxyyzz abc == is
111 ,, xxatyybtzzct =+=+=+ , where t is parameter.
iii. General point on the line
111 xxyyzz abc == is () 111 ,, xarybrzcr +++ , where || r is the distance between the general point and () 111 ,, xyz .
2. The equation of a line passing through the point () 111 ,, Pxyz and having direction cosines are ,, lmn is 111 xxyyzz lmn == .
The equation of a line passing through the origin and having direction ratios proportional to ,, abc is y xz abc ==
3. The equation of a line passing through two points () 111 ,, Pxyz and () 222 ,, Qxyz is
111 212121 xxyyzz xxzzzz == . The vector form of a line passing through the points , ab
is ()ratba =+−
.
4. To check if the given three points are collinear or not, find the equation of the line passing through any two points and then substitute the third point in the equation. If all three fractions give the same result, then those three points are collinear.
5. Equations of axes in symmetric form:
i. Equation of x-axis is . 100 y xz ==
ii. Equation of y-axis is . 010 y xz ==
iii. Equation of z-axis is 001 y xz ==
16. Find the equation of the line passing through the points () 1,0,2 and ()3,4,6.
Sol: Equation of a line passing through the points () 1,0,2 and () 3,4,6 is 12 444 y xz+− == This can be simplified as 12xyz+==−
Try yourself:
16. Find the coordinates of a point on the line 1 1 23 xyz + == at a distance 414 units from the point () 1,1,0.
Ans: )( 9,13,4 and )( 7,11,4
13.3.3 Reducing the General Form of a Line to Symmetric Form of a Line
General form of a line is 11112222 0. axbyczdaxbyczd +++==+++
To reduce this equation to symmetric form, consider the following steps.
Step I: Find 111 222
abc abc . The components of this vector are proportional to direction ratios of the given line.
Step II: If 1221 0 abab−≠ , then the given line intersects xy-plane, so that sub 0 z = in the general form of line. Solve the simultaneous equations 111222 0 axbydaxbyd ++==++ to get () , xy . Suppose, the point is () 11,,0xy .
Step III: Symmetric form is 11 122112211221 xxyyz bcbccacaabab == .
If 1221 0 abab−= , then take 0 y = and proceed as above.
17. Reduce the equation of the line 3240423 xyzxyz +−−==+−+ in symmetric form.
Sol: Let ,, abc be the direction ratios of the line which is the line of intersection of planes 3240 xyz+−−= and 4230 xyz+−+=
Hence, 320 abc+−= and 420 abc+−= .
By using cross multiply method, 325 abc ==
There is no zero in the direction ratios. The line is not parallel to any of the axes.
Let 0 z = , so that 324 xy+= and 43 xy+=− .
Solving these two equations, the point of intersection is () 2,5,0
Therefore, the equation of the line in symmetric form is 5 2 325 y xz + == .
Try yourself:
17. Write the equation of the line 2402 xzyz +−==+ in symmetric form :Ans 2 4 112 y xz + ==
13.3.4 Angle between Two Lines
Let θ be the acute angle between two lines having direction ratios proportional to 111 ,, abc and 222 ,, abc . Then, 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++
If θ is acute angle between two lines
111 111 xxyyzz abc == and 222 222 xxyyzz abc == , then 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++
The condition that these two lines are to be perpendicular is 121212 0 aabbcc++= . The condition that these two lines are to be parallel is 111 222 abc abc ==
To find the direction of a line with greatest slope:
let Π1, Π2 be two planes intersecting a line l1 Then, the line of greatest slope in Π1 is the line lying in the plane Π 1 and perpendicular to the line l1.
Let and ab be the vectors along the normals to the planes Π1 and Π2 , respectively. Then, the vector ()aab ×× will be along the line of greatest slope in Π1.
18. Find the angle between the lines 3 12 321 y xz++ == and 7 7 32 xyz + == ,
Sol: where 1111,2,1 abc=== and 2221,3,2 abc==−= .
Use the formula for finding the angle between two lines 362 1 cos 2 941194 −−+ θ==± ++++
Therefore, the angle between the given two lines is 3 π or 2 3 π .
Try yourself:
18. Find the value of k for which the lines 3 12 322 y xz k ++ == and 2 31 317 y xz k ++ == are perpendicular to each other. Ans: 2
13.3.5 Equations of Lines with the Given Conditions
The equation of a line passing through () 222 ,, xyz and parallel to the line 111 xxyyzz abc == is
222 xxyyzz abc ==
The equation of a line passing through p and perpendicular to the lines ratb =+ and rctd =+ is ()rptbd =+×
.
19. Find the equation of a line passing through () 1,2,3 and parallel to the line
1 4 10 238 y xz + −+ ==
Sol: Equation of required line is
222 , xxyyzz abc == where ()() ,,2,3,8abc =− and ()() 222,,1,2,3.xyz =−
Therefore, the equation of the line passing through the point (–1, 2, 3)and having direction ratios proportional to 2,3,8 is
2 13 238 y xz+− ==
Try yourself:
19. Find the equation of the line passing through () 2,3,4 and parallel to the planes 2345 xyz++= and 3456. xyz++=
Ans: 3 24 121 y xz+− ==
13.3.6 Coplanarity and Non-coplanarity of Two Lines
Two lines are said to be coplanar if the lines are either parallel or intersect each other. Two lines are said to be non-coplanar or skew lines if they are neither parallel nor intersecting each other. Two skew lines possess common perpendicular. The length of the common perpendicular is called the shortest distance between those lines.
The lines 111 111 xxyyzz abc == and 222 222 xxyyzz abc == are coplanar if and only if 212121 111 222 0 xxyyzz abc abc =
and the equation of this plane is 111 111 222 0 xxyyzz abc abc =
If the lines 111 xxyyzz lmn == and 11112222 0 axbyczdaxbyczd +++==+++ are coplanar, then 11111112121212 111 222 axbyczdaxbyczd albmcnalbmcn ++++++ = ++++
Point of Intersection of Two Lines
If two lines are coplanar, then those two lines are either parallel or intersect each other.
To find the point of intersection of two lines, consider the following steps:
1. Write the equations of the lines in parametric forms.
2. Equate the corresponding coordinates () ,, xyz of the lines.
3. Solve the equations to solve the parameters.
4. Substitute the parameter values into parametric equations of lines. If both lines give the same point, then it is the point of intersection of the given two lines.
20. Find the value of k if the lines 1 3 315 y xzk +− == and 2 15 125 y xz+− == are coplanar.
Sol: Points on the given lines are ()() 111,,3,1,5,xyz =− (x2, y2, z2)=(–1, 2, 5).
Direction rations of lines are ()()()() 111 222 ,,3,1,5,,,1,2,5 abcabc=−=
Since lines are coplanar, then 215 3150 125 k −=⇒ 5 k =
Try yourself:
20. Find the point of intersection of lines
3 41 147 y xz + −+ == and 1 1 10 . 238 y xz + −+ ==
Ans: )( 5,7,6
13.3.7 Perpendicular Distance from a Point to the Line
Let () 222 ,, Pxyz be any point in the space not on the line 111 xxyyzz abc == . To find the perpendicular distance from the point () 222 ,, Pxyz to the line 111 , xxyyzz abc == consider the following steps.
Step I: Take the general point Q on the line.
Step II: Find the direction ratios of .PQ
Step III: Since the given line and PQ are perpendicular, use the condition for
perpendicular lines. Find the parmeter.
Step IV: Substitute the parameter value in Q to get the foot of the perpendicular of P on the given line.
21. Find the perpendicular distance of the point () 2,1,4 from the line 2 3 . 1071 y xz + ==
Sol: Given line is 2 3 1071 y xz + == and the given point is () 2,1,4 P
Let any point on the line be () 103,72,. Q λ−−λ+λ
Direction ratios of PQ are 105,73,4. λ−−λ+λ−
The line PQ is perpendicular to the given line. Hence, ()()() 10105773140 10050492140 150750 1 2 λ−−−λ++λ−= λ−+λ−+λ−= λ−= λ=
The coordinates of the point Q are 31 2,, 22
The distance between P and Q is the required perpendicular distance from P to .Q Therefore, the length of the perpendicular is 52 2
Try yourself:
21. Find the length of the perpendicular from the point () 3,1,11 to the line 2 3 234 y xz ==
Ans: 53
13.3.8 Distance between Two Parallel Lines
The distance between two parallel lines 111 xxyyzz abc == and
222 xxyyzz abc == is 212121 222 ijk xxyyzz abc abc ++
The distance between two parallel lines
ratb =+ and rctb =+ is ()bac b ×−
22. Find the distance between the parallel lines ()() 2 rijijk =−+λ+− and ()() 2 rjkijk =−+µ+−
Sol: Where ,2, aijbijkcjk =−=+−=−
Now, 2 acijk −=−+ () 21135 121 ijk bacijk ×−=−=−−−
Required distance = () 192535 6 411 bac b ×−++ == ++
Try yourself:
22. Find the distance between the parallel lines ()232 rijkijk =+−+λ−+ and ()342 rijkijk =−+++µ−+
Ans: 22 2 3
13.3.9 Shortest Distance between Two Skew Lines
The shortest distance between two lines 111 111 xxyyzz abc == and 222 222 xxyyzz abc == is ()()() 212121 111 222 222 122112211221 . xxyyzz abc abc bcbccacaabab −+−+− ,
If the numerator is zero, then the lines are either parallel or intersect each other. The shortest distance between two lines
ratb =+
and rctd =+ is ()() acbd bd −⋅× ×
. If the numerator is zero, then the lines are either parallel or intersect each other.
If they are parallel, then find the distance between the lines using the previous procedure. If the lines intersect, then the distance between the lines is zero.
23. If the shortest distance between the lines 2 1 211 y xz−λ− == and 1 32 121 y xz == is 1, then find the sum of all possible values of λ .
Sol: Given lines are passing through the points () ,2,1 λ and () 3,1,2 .
Shortest distance = 311 211 121 211 121 ijk −λ− ()33 133 33 −λ ⇒=⇒−λ=±
0 ⇒λ= or 23 λ=
Sum of values of 23 λ=
Try yourself:
23. Find the shortest distance between the lines
()() 222 rijkijk =−−+λ++
and ()() 2 rijkijk =+++µ−+
TEST YOURSELF
Ans: 3 2
1. The equation of the line passing through two points (1, 2, 3) and (2, –1, 2) is
(1) 223 212 == xyz
(2) 123 131 == xyz
(3) 211 123 −+− == xyz
(4) 123 212 == xyz
2. Equation of a line passing through the point (1, –2, 5) and having direction ratios (1, 2, 3) is
(1) 125 123 −+− == xyz
(2) 125 123 +−+ == xyz
(3) 125 321 −+− == xyz
(4) 123 125 == xyz
3. Equation of the line passing through (–1, 2, 3) and parallel to 4110 238 −++ == xyz is
(1) 123 238 == xyz
(2) 123 238 +−− == xyz
(3) 4110 123 −++ == xyz
(4) 4110 123 −++ == xyz
4. The angle between two lines 123 212 −−+ == xyz and 110 == xyz is
(1) 0° (2) 30° (3) 45° (4) 90°
5. The foot of the perpendicular of the point (3, –1, 11) to the line 23 234 == xyz is
(1) (2, 5, 7) (2) (2, 3, 4) (3) (0, 2, 3) (4) (3, –1, 11)
6. The symmetric form of the line equation x–y+2 z = 5, 3x+y+z = 7 is
(1) 41149 331 −+ == xyz
(2) 119 331 −+ == xyz
(3) 4149 331 == xyz
(4) 32 354 −+ == xyz
7. The distance of the point (1, –5, 9) from the plane x – y + z = 5, measured along a straight line x = y = z, is ______.
8. If two lines L 1 : x = 5, 32 yz α = and L2 : x = α, 12 yz α = are coplanar, then α can take value(s) α 1, α 2. Then, α 1 + α 2 = _______, where ( α≠ 5).
9. If the line 324 213 −++ == xyz lies in the plane lx+my–z = 9, then l2+m2 is equal to _____.
Answer Key
(1) 2 (2) 1 (3) 2 (4) 3 (5) 1 (6) 4 (7) 17. 32 (8) 5 (9) 2
13. 4 LINE AND PLANE
13.4.1 Angle between Line and Plane
If the angle between the line and normal to the plane is θ , then the angle between the line and the plane is 90°−θ . If the angle between the line 111 xxyyzz lmn == and the plane 0 axbyczd+++= is θ , then 222222 sin albmcn abclmn θ=++ ++++ .
The condition for the line 111 xxyyzz lmn == to be perpendicular to the plane 0 axbyczd+++= is abc lmn == .
The condition for the line 111 xxyyzz lmn == to be parallel to the plane 0 axbyczd+++= is 0 albmcn++=
24. Find the angle between the line 2 13 212 y xz−+ == and the plane
40 xy++=
Sol: Given line is 2 13 212 y xz−+ == , where
()() ,,2,1,2lmn = and given plane is 40 xy++= , where 1,1,0.abc===
Now, ()() 12110 31 sin 110414322 ++ θ=== ++++ 4 ⇒θ=π
Try yourself:
24. If the angle between the line 1 3 2 xyz == λ and the plane 234xyz++= is
then find the value of λ
Ans: 0. 66
13.4.2 Relation between Line and Plane
The conditions for the line 111 xxyyzz lmn == to not intersect or not be parallel to the plane 0 axbyczd+++= are 0 albmcn++= and 111 0 axbyczd+++≠
The conditions for the line 111 xxyyzz lmn == to completely lie on the plane ax+by+cz+d=0 are al + bm + cn=0 and ax1 + by1+cz1 + d = 0.
The condition for the line 111 xxyyzz lmn ==
to intersect the plane 0 axbyczd+++= is 0 albmcn++≠ .
To find the point of intersection of the line 111 xxyyzz lmn == and the plane 0 axbyczd+++= , co nsider the following steps.
Step I: Take the general point on the line as () 111 ,, ltxmtyntz +++ , where t is parameter.
Step II: Since the point lies on the plane, the equation obtained by substituting the point in the equation of plane should be satisfied.
Step III: Solve the above equation for the parameter . t
Step IV: Substitute the value of t in the general point to get the point of intersection of line and plane.
25. If the plane ()()() 412850 xkyz −+−+−= contains the line 2 15 243 y xz == , then find the value of k
Sol: The given plane is 484420 xkyzk ++−−= . If the line lies on the plane, then 0 albmcn++= and 111 0 axbyczd+++= . Hence, ()()() 424830 260 8 k k k ++= ++= =− Therefore, the value of |k| is 8.
Try yourself:
25. Find the distance of the origin from the point of intersection of line 2 3 234 y xz == and the plane 22. xyz+−=
Ans: 78
TEST YOURSELF
1. The equation of a line passing through the point (–1, 2, –3) and perpendicular to the plane 2x+3y+z+5 = 0 is
(1)
(2)
(3)
(4)
123 111 −+− == xyz
123 111 +−+ == xyz
123 231 +−+ == xyz
123 221 +−+ == xyz
2. If θ is the angle between a line 112 122 +−− == xyz and the plane 2x–y+ λ z + 4 = 0 is such that sin θ = 1 3 , then the value of λ is (1) 3 4 (2) 4 3 (3) 5 3 (4) 3 5
3. The line 321 321 +−+ == xyz and the plane
4x+5y+3z–5 = 0 intersect at a point (1) (3, 1, –2) (2) (3, –2, 1) (3) (2, –1, 3) (4) (–1, –2, –3)
4. Let the line 212 352 −−+ == xyz lie in the plane x+3y– α z+ β = 0. Then, ( α , β ) equals (1) (–6, 7) (2) (5, –15) (3) (–5, 5) (4) (6, –17)
5. The distance between the line r = 2i–2 j+3 k+λ(i–j+4k) and the plane r .(i+5j+k) = 5 is (1) 3 10 (2) 10 3 (3) 10 9 (4) 10 33
6. The angle between the line 123 212 −−+ == xyz and the plane x+y+4 = 0 is (1) 0 (2) 6 π (3) 4 π (4) 2 π
7. The distance of the point (1, 0, –3) from the plane x–y–z = 9, measured parallel to the line 226 236 −+− == xyz , is _____.
8. A line with direction ratios (2, 7, –5) is drawn to intersect the lines 7 52 311 y xz−+ == and 336 324 +−− == xyz at P and Q, respectively. Length of (PQ)2 is _____.
9. If the angle between the line 13 2 == yz x λ and the plane x+2y+3z = 4 is 1 5 cos 14 , then λ = ____.
Answer Key (1) 3 (2) 3 (3) 2 (4) 4 (5) 4 (6) 3 (7) 7 (8) 78 (9) 0. 66
CHAPTER REVIEW
Direction Cosines and Direction Ratios
1. If ,, lmn are direction cosines of a line L , then ,, lmn are also direction cosines of the same line L
2. If ,, lmn are direction cosines of a line L , then 222 1. lmn++=
3. If ,, αβγ are the angles made by a line L with positive direction of axes, then
• 222 coscoscos1 α+β+γ=
• 222 sinsinsin2 α+β+γ=
• cos2cos2cos21 α+β+γ=−
4. Direction cosines of a line which makes equal angles with coordinate axes are 111 ,, 333 ±±± .
5. If ,, lmn are direction cosines of a line, then the maximum value of lmn is 1 33
6. The direction ratios of a line joining the points () 111 ,, Axyz and () 222 ,, Bxyz are 212121,,. xxyyzz
7. Suppose, the direction cosines of two lines are 111 ,, lmn and 222 ,, lmn . If θ is the acute angle between the above two lines, then 121212 cos llmmnnθ=++ .
8. The condition that the lines are to be perpendicular is 121212 0. llmmnn++= The condition that the lines are to be parallel is 111 222 lmn lmn ==
9. Suppose, the direction ratios of two lines are 111 ,, abc and 222 ,, abc . If θ is the acute angle between the above two lines, then 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++
10. The condition that the lines are to be perpendicular is 121212 0 aabbcc++= . The condition that the lines are to be parallel is 111 222 abc abc == .
11. If a ray makes angles ,,, αβγδ with the four diagonals of a cube, then
2222 4 coscoscoscos. 3 α+β+γ+δ=
12. If 111 ,, lmn and 222 ,, lmn are direction cosines of two intersecting lines, then the direction ratios of angular bisectors of the above two lines are 121212 ,, llmmnn ±±± .
13. If 111 ,, lmn and 222 ,, lmn are direction cosines of two lines then the direction cosines of angular bisectors of the above two lines are 121212 ,, 2cos2cos2cos 222 llmmnn +++ θθθ and 121212 ,, 2sin2sin2sin 222 llmmnn θθθ , where θ is an angle between the lines.
14. If 111 ,, lmn , 222 ,, lmn , and 333 ,, lmn are the direction cosines of three mutually perpendicular lines, then direction ratios of a line which makes equal angles with given three lines are 123123123 ,,. lllmmmnnn ++++++
15. If 111 ,, lmn and 222 ,, lmn are direction cosines of two lines, then the direction ratios of a line which is perpendicular to both the given lines are 122112211221 ,,.mnmnnlnllmlm
16. The lines whose direction cosines are related with relations 0 albmcn++= and 0 fmngnlhlm++= are
• perpendicular if 0 fgh abc ++= ,
• parallel if 222222 2220afbgchabfgbcghacfh ++−−−= (or) 0 afbgch±±= .
17. The lines whose direction cosines are related with relations 0 albmcn++= and 222 0 ulvmwn++= are
• perpendicular if ()()() 222 0 avwbuwcuv+++++= ,
• parallel if 222 0 abc uvw ++=
18. If the edges of a rectangular parallelopiped are ,, abc then the angle between its diagonals are 222 1 222 cos abc abc ±±±
. (In the numerator all the three signs are not either positive or negative.)
19. Angle between the diagonals of a cube is 1 1 cos 3
20. The angle between the a diagonal of a cube and the diagonal of a face of the cube is 1 2 cos 3
.
21. If d is the length of the line segment and 123 ,, ddd are the projections made by the line segment on the axes, then
2222 123 dddd =++
22. If d is the length of the line segment and 123 ,, ddd are the projections of the line segment on coordinate planes, then 2222 123 2. dddd =++
Plane in 3D Geometry
1. The general equation of a plane is 0. axbyczd+++= Here, ()() ,,0,0,0.abc ≠ The ordered triad ,, abc are direction ratios of normal to the plane. Here, normal is a perpendicular line to the plane.
2. The equation of any plane passing through the point () 111 ,, xyz and having direction ratios of normal to the plane ,, abc is ()()() 111 0. axxbyyczz−+−+−=
3. The equation of a plane passing through three points ()()() 111222333 ,,,,,,,, AxyzBxyzCxyz is 111 212121 313131 0 xxyyzz xxyyzz xxyyzz −−−= . The components of the vector ABAC ×
are the direction ratios of normal to the plane containing the points ,, ABC . There exists unique plane passing through three noncollinear points.
4. If the points ()()() 111222333 ,,,,,,,,, xyzxyzxyz and () 444 ,, xyz are collinear, then 414141 212121 313131 0 xxyyzz xxyyzz xxyyzz −−−=
5. The equation of a plane, which is at a distance of p units from the origin and whose normal has direction cosines ,, lmn , is lxmynzp ++=
6. The equation of a plane passing through the origin and whose normal has direction cosines ,, lmn is 0. lxmynz++=
7. The equation of a plane passing through the point () 111 ,, Pxyz and having normal line, whose direction ratios are ,, abc , is ()()() 111 0. axxbyyczz−+−+−=
8. The normal form of the plane 0 axbyczd+++= , is
2222 , abcd xyz aaaa ++=− ∑∑∑∑
when 0 d < .
9. The normal form of the plane 0 axbyczd+++= is 2222 , abcd xyz aaaa −−−=
when 0. d >
10. The equation of the plane having intercepts ,,,abc respectively, is 1 y xz abc ++= .
11. The intercept form of the plane
0 axbyczd+++= is 1. y xz ddd abc ++=
12. The equation of plane whose intercepts are k times the intercepts made by the plane 0 axbyczd+++= on corresponding axes is 0. axbyczkd+++=
13. If a plane meets the coordinate axes at ,, ABC and the centroid of the triangle ABC is () ,, pqr , then the equation of the
plane is 3 y xz pqr ++= .
14. The area of the triangle formed by th e
plane 1 y xz abc ++= , with
•x-axis and y-axis is 1 2 ab square units
•y-axis and z-axis is 1 2 bc square units.
•x-axis and z-axis is 1 2 ac square units.
15. If the plane 1 y xz abc ++= cuts the axes at ,, ABC then the area of the triangle formed by ,, ABC is ()()() 222 1 2 abbcca ++
16. The perpendicular distance from the origin to the plane 0 axbyczd+++= is 222 d abc ++
17. The perpendicular distance from () 111 ,, Pxyz to the plane 0 axbyczd+++= is 111 222 axbyczd abc +++ ++
18. Two points () 111 ,, Axyz and () 222 ,, Bxyz lie on the same side of the plane
0. axbyczd+++= If the values 111 axbyczd +++ and 222 axbyczd +++ have the same sign.
19. Two points () 111 ,, Axyz and () 222 ,, Bxyz lie on opposite sides of the plane 0 axbyczd+++= if the values 111 axbyczd +++ and 222 axbyczd +++ have opposite signs.
20. The ratio in which the plane 0 axbyczd π≡+++= divides the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz is –π111:π222 ,i.e. , 111 222 . axbyczd axbyczd +++ +++
21. If () ,, Qhkl is foot of the perpendicular of () 111 ,, Pxyz on the plane 0 axbyczd+++= , then () 111 111 222 . hxkylzaxbyczd abcabc −+++ === ++ .
22. If () ,, Qhkl is image of the point () 111 ,, Pxyz with respect to the plane 0 axbyczd+++= , then () 111 111 222 2 hxkylzaxbyczd abcabc −+++ === ++
23. The reflection of the plane 0 axbyczd ′′′′+++= in the plane 0 axbyczd+++= is given by () () () () 222 2 aabbccaxbyczd abcaxbyczd ′′′ +++++ ′′′′=+++++
24. Let θ be the acute angle between two planes 1111 0 axbyczd+++= and 2222 0 axbyczd+++= . Then 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++
25. The condition that the two planes are to be perpendicular to each other is 121212 0 aabbcc++= . The condition that the two planes are to be parallel to each other is 111 222 abc abc == . If the angle between two planes is 90° , then those planes are called orthogonal planes.
26. The equation of any plane parallel to the plane 0 axbyczd+++= can be taken as 0. axbyczk+++=
27. The equation of plane passing through the point () 111 ,, Pxyz and parallel to the plane 0 axbyczd+++= is ()()() 111 0. axxbyyczz−+−+−=
28. The distance between two parallel planes 1 0 axbyczd+++= and 2 0 axbyczd+++= is 21 222 dd abc ++
29. The equation of plane which is midway between two parallel planes 1 0 axbyczd+++= and 2 0 axbyczd+++= is 12 0. 2 dd axbycz +
+++=
30. The equation of any plane passing through the line of intersection of two planes 1π and 2π can be taken as 12 0 π+λπ= .
31. The equation of any plane passing through () 111 ,, Pxyz and parallel to the lines whose direction ratios are 111 ,, abc and 222 ,, abc is 111 111 222 0 xxyyzz abc abc =
32. The equation of any plane passing through ()() 111222 ,,,,, PxyzQxyz and parallel to the line whose direction ratios are ,, abc is 111 212121 0 xxyyzz xxyyzz abc −−−=
33. Equation of plane which is perpendicular bisector of the line segment joining () 111 ,, Axyz and () 222 ,, Bxyz is a plane passing through the midpoint of AB and having the line AB as normal.
34. Equation of two planes bisecting the angles between the planes 1111 0 axbyczd+++= and 2222 0 axbyczd+++= are 11112222 222 222 111 222 axbyczdaxbyczd abcabc ++++++ =± ++++
35. Suppose, 12,0dd > . Then, the equation 11112222 222222 111222 axbyczdaxbyczd abcabc ++++++ = ++++ represents acute angular bisector, if 121212 0 aabbcc++< .
36. Suppose, 12,0dd > . Then, the equation 11112222 222222 111222 axbyczdaxbyczd abcabc ++++++ = ++++ represents obtuse angular bisector, if 121212 0 aabbcc++>
Straight line in space
38. The equation of a line passing through the point () 111 ,, Pxyz and having direction ratios proportional to ,, abc is 111 xxyyzz abc == . This is called symmetric form of a line.
39. Vector form of the line 111 xxyyzz abc == is
() 111 rxiyjzktaibjck =+++++
. This
vector form of a line passes through the point whose position vector is 111 xiyjzk ++ and parallel to the vector aibjck ++ .
40. Parametric form of line 111 xxyyzz abc == is 111,,, xxatyybtzzct =+=+=+ where t is parameter.
41. General point on the line 111 xxyyzz abc == is () 111 ,, xarybrzcr +++ , where r is distance between the general point and () 111 ,, xyz
42. The equation of a line passing through the point () 111 ,, Pxyz and having direction cosines ,, lmn is 111 xxyyzz lmn == .
43. The equation of a line passing through the origin and having direction ratios proportional to ,, abc is y xz abc == .
44. The equation of a line passing through two points () 111 ,, Pxyz and () 222 ,, Qxyz is 111 212121 , xxyyzz xxzzzz = . The vector form of a line passing through the points and ab is ()ratba =+−
.
45. Let θ be the acute angle between two lines having direction ratios proportional to 111 ,, abc and 222 ,, abc then 121212 222222 111222 cos aabbcc abcabc ++ θ= ++++ ,
46. The condition that these two lines are to be perpendicular is 121212 0 aabbcc++= . The condition that these two lines are to be parallel is 111 222 abc abc ==
47. The equation of a line passing through () 222 ,, xyz and parallel to the line 111 xxyyzz abc == is 222 xxyyzz abc ==
48. The equation of a line passing through p and perpendicular to the lines ratb =+ and rctd =+ is ()rptbd =+×
49. The lines 111 111 xxyyzz abc == and 222 222 xxyyzz abc == are coplanar if and only if 212121 111 222 0, xxyyzz abc abc = and the equation of this plane is 111 111 222 0 xxyyzz abc abc =
50. The distance between two parallel lines 111 xxyyzz abc == and 222 xxyyzz abc == is 212121 222 ijk xxyyzz abc abc ++
51. The distance between two parallel lines ratb =+ and rctb =+ is ()bac b ×−
52. The shortest distance between two lines 111 111 xxyyzz abc == and 222 222 xxyyzz abc == is ()()() 212121 111 222 222 122112211221 . xxyyzz abc abc bcbccacaabab −+−+−
If the numerator is zero, then the lines are either parallel or intersect each other.
53. The shortest distance between two lines ratb =+ and rctd =+ is ()() . acbd bd −⋅× ×
If the numerator is zero, then the lines are either parallel or intersect each other.
Line and Plane
54. If the angle between the line and normal to the plane is θ , then the angle between a line and the plane is 90°−θ . If the angle between a line 111 xxyyzz lmn == and the plane 0 axbyczd+++= is θ , then 222222 sin albmcn abclmn θ=++ ++++
CHAPTER 13: 3D Geometry
55.The condition for the line
111 xxyyzz lmn == is to be perpendicular to the plane 0 axbyczd+++= is abc lmn == . The condition for the line
111 xxyyzz lmn == is to be parallel to the plane 0 axbyczd+++= is 0 albmcn++= .
57. The conditions for the line 111 xxyyzz lmn == completely lies on the plane 0 axbyczd+++= are 0 albmcn++= and 111 0 axbyczd+++=
58. The condition for the line 111 xxyyzz lmn == intersects the plane 0 axbyczd+++= is
56. The conditions for the line 111 xxyyzz lmn == do not intersect or parallel to the plane 0 axbyczd+++= are 0 albmcn++= and 111 0 axbyczd+++≠
0 albmcn++≠
Exercises
JEE MAIN LEVEL
Level – I
Direction Cosines and Direction Ratios
Single Option Correct MCQs
1. The vertices of a triangle are (2, 3, 5), (–1, 3, 2), (3, 5, –2), then the angles are (1) 30°, 30°, 120°
(2)
(3) 30°, 60°, 90° (4) 1 1 cos 3
2. The angle between a diagonal of a cube and the diagonal of a face of the cube is
6. The acute angle between the two lines whose dc's are given by l+m–n = 0 and l2+m2 –n2 = 0 is
7. If the direction cosines of two lines are given by l+m+n = 0 and l2–5m2+n2 = 0, then the angle (1) 6
8. If the direction ratios of two lines are given by 3lm – 4ln+ mn = 0 and l + 2m + 3n = 0, then the angle between the lines is (1) 2
9. If (1, 2, 1), (1, –3, 2) are the direction ratios of two lines and (l, m, n) are the direction cosines of a line perpendicular to the given lines, then l + m + n= (1) 1 (2) 1
3. A(–1, 2, –3), B(5, 0, –6) and C(0, 4, –1) are the vertices of a triangle. The dr's of the internal bisector of ∠ BAC are (1) (25, –8, –5) (2) (5, 6, 8) (3) (25, 8, 5) (4) (4, 7, 9)
4. The distance of the point A(–2, 3, 1) from the line PQ through P(–3, 5, 2) which make equal angles with axes, is
5. Angle between a diagonal of a cube and edge of cube is
10. The cosine of the angle A of the triangle with vertices A (1, –1, 2), B (6, 11, 2), C(1, 2, 6) is (1) 63 65 (2) 36 65 (3) 16 65 (4) 13 64 Numerical Value Questions
11. A line makes angles α, β, γ with the positive directions of the axes. Then |cos 2 α + cos 2 β + cos 2 γ |is _____.
12. The number of lines which are equally inclined to the coordinate axes is _____.
13. The projections of a line segment on the axes are 3, 4, 12 then the length of the line segment is ______.
14. If A = (–2, 3, 4), B = (–4, 4, 6), C = (4, 3, 5), D = (0, 1, 2) then the projection of AB on CD is ______.
15. The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, 3) and (3, –2, 10) is ______.
Plane in 3D Geometry
Single Option Correct MCQs
16. The equation of plane passing through a point A(2, –1, 3) and parallel to the vectors a = (3, 0, –1) and b = (–3, 2, 2) is
(1) 2x–3y+6z–25 = 0
(2) 2x–3y+6z+25 = 0
(3) 3x–2y+6–25 = 0
(4) 3x–2y+6+25 = 0
17. The equation of the plane parallel to the plane 2 x + 3 y + 4 z + 5 = 0 and passing through the point (1, 1, 1) is
(1) 2x+3y+4z–9 = 0
(2) 2x+3y+4z+9 = 0
(3) 2x+3y+4z+7 = 0
(4) 2x+3y+4z–7 = 0
18. A plane meets the coordinate axes at A, B, C so that the centroid of the triangle ABC is (1, 2, 4). Then the equation of the plane is
(1) x+2y+4z = 12 (2) 4x+2y+z = 12
(3) x+2y+4z = 3 (4) 4x+2 +z = 3
19. The foot of the perpendicular from (1, 3, 4) to 2x–y+z+3=0, is
(1) (1, –4, 3) (2) (–1, 4, 3) (3) (0, 3, 0) (4) (1, 2, 3)
20. The image of the point (5, 2, 6) with respect to the plane x + y + z = 9 is
(1) (3, –5, 2) (2) 7 ,1,5 2
(3) 7210 ,, 333
(4) 5 72 ,, 333
21. A plane containing the point (3, 2, 0) and the line 123 154 == xyz and also contains the point (1) (0, –3, 1) (2) (0, 7, 10) (3) (0, 7, –10) (4) (0, 3, 1)
22. A point on the plane passes through the points (1, –1, 6), (0, 0, 7) and perpendicular to the plane x–2y+z = 6 is (1) (1, –1, 2) (2) (1, 1, 2) (3) (–1, 1, 2) (4) (1, 1, –2)
23. A plane passes through the point (3, 5, 7). If the direction ratios of its normal are equal to the intercepts made by the plane x+3y+2z = 9 with the coordinate axes, then the equation of that plane is
(1) x+y+z = 5
(2) 6x+2y+3z = 105
(3) 12x+4y+6z = 49
(4) 6x+2y+3z = 49
24. The equation of the plane containing the line 2x–5y+z = 3, x+y+4z = 5 and parallel to the plane, x+3y+6z = 1, is
(1) 2x+6y+12z = 13 (2) x+3y+6z = –7
(3) x+3y+6z = 7 (4) 2x+6y+12z = –13
25. The equation of the plane containing the lines 2 x –5 y + z = 3, x + y +4 z = 5 and perpendicular to the plane x+3y+6z = 1, is
(1) 2x+6y+12z = 13
(2) x+3y+6z = –7
(3) 9x–19y+8z = 17
(4) 2x+6y+12z = –13
26. Equation of a plane at a distance 2 21 from the origin, which contains the line of intersection of the planes x–y–z–1 = 0 and 2x+y–3z+4 = 0, is
(1) 3x–4z+3 = 0 (2) 4x–y–5z+2 = 0
(3) –x+2y+2z–3 = 0 (4) 3x–y–5z+2 = 0
Numerical Value Questions
27. The angle between the planes kx+y–z = 9 and x+2y+z = 7 is 3 π then k = _____
28. P = (0, 1, 0), Q = (0, 0, 1) then the square of projection of PQ on the plane x+y+z = 3 is ______.
29. Distance between two parallel planes 2 x + y + 2z = 8 and 4x+ 2y + 4z +5 = 0 is _____.
30. A line with positive direction cosines passes through the point P (2,–1,2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q . The length of the line segment PQ =
31. The plane x – y – z = 2 is rotated through an angle 90° about its line of intersection with the plane x + 2y + z =2. Then equation of this plane in new position is 5 x + 4y + z = k then 5k = ______.
32. If the angle between the planes 2x+y+2z+3 = 0, x–y+kz–2 = 0 is 3 4 π then k = ______.
33. The ratio in which the line segment joining the points whose position vectors are ˆ 27 ˆ 4 ˆ ijk and 8 ˆ ˆ ˆ 35ijk−+− is divided by the plane whose equation i s () 3 ˆ ˆ ˆ ˆ 231rijk⋅−+= is m : n, then | m+n | = ______.
Straight Line in Space
Single Option Correct MCQs
34. The value of k for which the lines 154 322 ++− == xyz k and 321 317 +−+ == xyz k are perpendicular, is (1) 3 (2) –1 (3) 2 (4) –2
35. If the lines 123 24 == xyz a and 125 12 ++− == xyz b are parallel, then a + b = (1) 3 (2) 1 (3) 5 (4) 2
36. If two lines, whose equations are 2 31 23 y xz λ == and 3 22 333 y xz == lie in the same plane. Then, the value of sin–1(sin λ ) is (1) 3 (2) π – 3 (3) 4 (4) π – 4
37. The angle between the lines 2x = 3y = –z and 6x = –y = –4z is (1) 45° (2) 30° (3) 0° (4) 90°
38. Angle between the lines 21 , 2 32 −+ == xy z and 3 15 2 334 y xz + −+ == is (1) 2 π (2) 3 π (3) 6 π (4) 4 π
39. The line x = 1, y = 2 is (1) parallel to the x-axis (2) parallel to the y-axis (3) parallel to the z-axis (4) parallel to the xy-plane
40. Let the line 212 352 xyz −−+ == lie in the plane x+3y– α z+ β = 0, then ( α , β ) = (1) (–6, 7) (2) (5, –15) (3) (–5, 5) (4) (6, –17)
41. A plane which passes through the point (3, 2, 0) and the line 474 154 xyz == is (1) x–y+z = 1
(2) x+y+z = 5
(3) x+2y–2 = 1 (4) 2x–y+z = 5
42. The lines 111 234 xyz−+− == and 3 121 xykz == intersect, if k equals to (1) 3 2 (2) 9 2 (3) 2 9 (4) 3 2
43. The lines 323 12 xyz l == and 121 34 xyz l == are coplanar if the value of l is (1) 2 (2) 13 (3) –13 (4) –11
44. The lines 234 11 xyz k == and 145 21 xyz k == are coplanar, if (1) k = 0 or –1 (2) k = 1 or –1 (3) k = 1 or –3 (4) k = 3 or –3
45. If the straight lines 123 23 xyz k == and 3 21 32 y xz k == intersect at a point, then the integer k is equal to (1) –5 (2) 5 (3) 2 (4) –2
46. The shortest distance between the lines 123 234 xyz == and 245 345 xyz == is (1) 1 6 (2) 1 6 (3) 1 3 (4) 1 3
47. Shortest distance between the lines 111 111 xyz == and 234 111 xyz == is ______.
Line and Plane
Single Option Correct MCQs
49. The planes bx–ay = n, cy–bz = l, az–cx = m intersect a line,if
(1) al–bm+cn = 1 (2) al+bm+cn = 0 (3) al+bm+cn+1 = 0 (4) al+bm+cn = 1
50. The equation of plane containing the line 161 342 xyz −++ == and parallel to the line
214 235 xyz −−+ == is
(1) 26x–11y–17z+109 = 0 (2) 26x+11y–17z–109 = 0
(3) 26x–11y–17z–109 = 0 (4) 26x–11y–17z+109 = 0
51. The point of intersection of the line 123 342 xyz −+− == and the plane 2x–
y+3z–1 = 0, is (1) (10, –10, 3) (2) (10, 10, –3)
(3) (–10, 10, 3) (4) (–10, –10, –3)
52. A variable plane passes through a fixed point (1, 2, 3). Then the foot of the perpendicualr from the origin to the palne lies on (1) a circle (2) a sphere (3) an ellipse (4) a parabola
53. The line 321 321 xyz +−+ == and the plane
4x + 5y + 3z – 5 = 0 intersect at a point (1) (3, 1, –2) (2) (3, –2, 1) (3) (2, –1, 3) (4) (–1, –2, –3)
Numerical Value Questions
48. If the line 111 234 xyz−+− == and 3 121 xykz == intersect, then k is equal to ______.
54. If the planes () 1 ˆ ˆˆ rijkq ⋅++= , () 2 ˆ ˆˆ 2 riajkq ⋅++= and () 2 3 ˆ ˆˆ raiajkq ⋅++=
intersect in a line, then the sum of all values of 'a' is _____
55. The value of k such that 42 112 xyzk == lies in the plane 2x–4y+z = 7, is ______.
56. Let the line 212 352 xyz −−+ == lie in the plane x+3y–αz+ β = 0. Then α2+ β 2 = ____.
57. The distance between the line () ˆˆˆˆˆ 2234 ˆ rijkijk λ =−++−+ and the plane () ˆ ˆˆ 55rijk⋅++= is ______.
Level – II
Direction Cosines and Direction Ratios
Single Option Correct MCQs
1. A straight line passing through (1, 1, 1) makes an angle of 60° with the positive direction of the z-axis and the cosines of the angles made with the positive directions of y , x -axis are in the ratio 3 : 1 then the acute angle between the two possible positions of the line is (1) 2 π (2) 6 π (3) 3 π (4) 4 π
2. If a line makes angles α, β, γ with positive axes, then the range of sinαsinβ + sinβsinγ + sinγsinα is (1) 1 ,1 2
(2) 1 ,2 2
(3) (–1, 2) (4) [–1, 2]
3. If the angle between lines with dc's 2 ,, 212121 ab and other line with dc's 336 ,, 545454
is 90° then a pair of possible values of a and b respectively are (1) –1, 4 (2) 4, 2 (3) 4, 1 (4) –4, –2
4. Let α be the angle between the lines whose direction cosines satisfy the equations l+ m–n = 0 and l2 + m2 – n2 = 0. Then the value of sin4α +cos4α is:
(1) 1 2 (2) 5 8 (3) 3 4 (4) 3 8
Numerical Value Questions
5. If the straight line whose dc's l,m,n are given by l + m = λn and mn+nl+lm = 0 be at right angles then λ = _____.
Plane in
3D Geometry
Single Option Correct MCQs
6. Equation of the plane through the midpoint of the join of A(4, 5, –10) and B(–1, 2, 1) and perpendicular to AB is
(1) 5x+3y–11z+ 135 2 = 0
(2) 5x+3y–11z = 135 2
(3) 5x+3y–11z = 135
(4) 5x+3y–11z+ 185 2 = 0
7. The equation of the bisectors of the angles between the planes 2x–y–2z–6 = 0, 3x+2y–6 z–12 = 0 is
(1) 23x–y–32z–78 = 0; 5x–13y–4z–6 = 0
(2) 15x–27y+2z–78 = 0; x–5y–15z–15 = 0
(3) 67x–162y+47z+44 = 0; 11x+6y+5z+86 = 0
(4) y–1 = 0, x+z–3 = 0
8. 5, 7 are the intercepts of a plane on the y -axis, z -axis respectively, if the plane is parallel to the x-axis then the equation of that plane is
(1) 5y +7z = 35 (2) 7y+5z = 1
(3) 35 57 +=yz (4) 7y+5z = 35
9. The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x+y–2z = 5 and 2x–5y–z = 7, is:
(1) 6x–5y+2z+10 = 0
(2) 3x–10y–2z+11 = 0
(3) 11x+y+17z+38 = 0
(4) 6x–5y–2z–2 = 0
10. A plane meets the coordinate axes in P, Q, R respectively. If the centroid of ∆ PQR is 11 1,,, 23
then the equation of the plane is
(1) 2x+4y+3z = 5 (2) x+2y+3z = 3
(3) x+4y+6z = 5 (4) 2x–2y+6z = 3
11. An equation of a plane parallel to the plane x–2y+2z–5 = 0 and at a unit distance from the origin is
(1) x–2y+2z–7 = 0 (2) x–2y+2z+5 = 0
(3) x–2y+2 –3 = 0 (4) x–2y+2z+1 = 0
12. The equation of the plane π through the line of intersection of the planes π 1 = x+3y–6 = 0 and 2π = 3x–y+4z = 0 is 12 0 +=πλπ . If the plane π is at unit distance form the origin, then an equation of the plane π is
(1) 2x+y+2z–3 = 0 (2) 2x–y–2z+3 = 0
(3) 2x+y+2z+3 = 0 (4) x+2y+2z+3 = 0
13. For the line 123 , 123 == xyz , which one of the following is incorrect
(1) It lies in the plane x–2y+z = 0
(2) It is same as line 123 == xyz
(3) It passes through (2, 3, 5)
(4) It is parallel to the plane x–2y+z–6 = 0
Numerical Value Questions
14. If a plane passing through (1, –2, 1) and is perpendicular to the planes 2x–2y+z = 0 and x – y +2 z = 4 then the square of the distance of the plane from (1, 2, 2) is ____.
15. The sum of the intercepts on the coordinate axes of the plane passing through the point (–2, –2, 2) and containing the line joining the points (1, –1, 2) and (1, 1, 1) is k then |k| = ______.
16. A line with positive direction cosines passes through the point P (2,–1,2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q . The length of the line segment PQ =
17. A line with positive direction cosines passes through the point P (2, –1, 2) and makes equal angles with the coordinate axes. If the line meets the plane 2 x+y+z = 9 at the point Q, then PQ2 equals _____.
Straight Line in Space
Single Option Correct MCQs
18. The length of the perpendicular drawn from the point (3, –1, 11) to the line 23 234 xyz == is (1) 66 (2) 29 (3) 33 (4) 53
19. The image of the line 134 315 xyz == in the plane 2x–y+z+3 = 0 is the line (1) 352 315 xyz −+− == (2) 352 315 xyz −+− == (3) 352 315 xyz +−− == (4) 352 315 xyz +−+ ==
20. Consider the lines 1 13 : 211 y xz L −+ == 2 433 : 112 xyz L −++ == and the planes
P 1 : 7 x + y +2 z = 3, P 2 : 3 x +5 y –6 z = 4. Let ax + by + cz = d the equation of the plane passing through the point of intersection of lines L1 and L2 and perpendicular to planes P1 and P2, then (a+d)–(b+c) = _____.
21. If the lines 234 11 xyz λ == and 145 21 xyz λ == intersect and λ take two values λ1, λ2, then λ1+λ2+10 = ______.
22. A line through origin meets the lines 23 122 xyz == and 1 224 xyz == at P and Q then (PQ)2 equals to _____.
23. The distance between the point (–1, –5, –10) and the point of intersection of the line 1 22 3412 y xz + == with the plane x–y+z = 5 is ______.
24. The point on the line 2634 2310 xyz ++− == which is nearest to the line 677 432 xyz +−− == is (a, b, c), then the value of a + b + c = _____.
Line and Plane
Single Option Correct MCQs
25. If the planes x = cy+bz, y = az+cx and z = bx+ay passes through the line if
(1) a2+b2+c2+2abc = 0 (2) a2+b2+c2+2abc = 1 (3) a2+b2+c2 = 2abc (4) a+b+c = abc
26. The lines () 2 rijik λ =−++ and ()() 2 rijijk µ =−++− intersect for (1) λ = 1, μ = 1 (2) λ = 2, μ = 3 (3) all value of λ and μ (4) no value of λ and μ
27. Let P be the image of the point (3, 1, 7) with respect to the plane x–y+z = 3. Then the equation of the plane passing through P and containing the straight line 121 xyz == (1) x+y–3z = 0 (2) 3x+z = 0 (3) x–4y+7z = 0 (4) 2x–y =0
28. The distance of point A(–2, 3, 1) from the line PQ through P(–3, 5, 2) which makes equal angles with the axes is (1) 2 3 (2) 14 3 (3) 16 3 (4) 5 3
29. Perpendiculars are drawn from points on the line 21 213 xyz ++ == to the plane x+ y+z = 3. The feet of perpendiculars lie on the line (1) 3 5 2 1 8 1 xyz == (2) 12 235 xyz == (3) 12 437 xyz == (4) 12 275 xyz ==
Numerical Value Questions
30. The distance from P(1, 1, 2) to the plane x+2y+z–1 = 0 measured parallel to the line 1 212 xyz + == is ______.
31. If θ is the angle between the line 112 324 xyz+−− == a nd the plane 2 x + y –3z+4 = 0, then () 2 8 cos 29 ec θ = ______.
32. The minimum distance of the point (1, 1, 1) from x+y+z = 1 measured perpendicular to the line 111 123 xxyyzz == is k, then the value of 9 ____. 21 = k
33. If the lines 461 352 xyz ++− == and 3x–2y+z+5 = 0 = 2x+3y+4z–k are coplanar, then the value of k = ______.
34. If the distance betwen the plane Ax–2y+z = d and the plane containing the lines 123 234 xyz == and 234 345 xyz == is 6 then |d| is ____.
35. The line 225 : 132 xyz L −+− == intersect the plane 2 x–3y+4z = 163 at point P and intersect the yz -plane at point Q . If the length of PQ is ab where a,b∈N and a>3, then the value of (b–a) is equal to ______.
Level – III
Single Option Correct MCQs
1. The equation of the plane in normal form which passes through the points (–2, 1, 3),(1, 1, 1) and (2, 3, 4) is (1) 2211 3333
(2) x–3y+2z–15 = 0
(3) 4x+3y–2z+15 = 0
(4) 4x+3y+2z+15 = 0
3. A variable plane is at a constant distance 3p from the origin and meets the axes A, B and C. The locus of the centroid of the triangle ABC is
(1) x–2+y–2+z–2 = p–2
(2) x–2+y–2+z–2 = 4p–2
(3) x–2+y–2+z–2 = 10p–2
(4) x2+y2+z2 = p2
4. A variable plane is at a constant distance p from the origin and meets the axes in A,B and C . The locus of the centroid of tetrahedron OABC is
(1) x–2+y–2+z–2 = p–2
(2) x–2+y–2+z–2 = 4p–2
(3) x–2+y–2+z–2 = 16p–2
(4) x2+y2+z2 = p2
5. The equation to the plane which passes through the z-axis and is perpendicular to the line 123 cossin0 xyz αα −+− == is
(1) xsinα+ycosα = 0
(2) xsinα–ycosα = 0
(3) xcosα+ysinα = 0
(4) xcosα–ysinα = 0
6. Equation of a line in the plane π≡ 2 x –y + z –4 = 0 which is perpendicular to the line l whose equation is 223 112 xyz == and which passes through the point of intersection of l and π is
(1) 211 151 xyz ==
xyz
(4) 41161 173173173173
2. The reflection of the plane 2 x–3y+4z–3 = 0 in the plane x–y+z–3 = 0 is the plane
(1) 4x–3y+2z–15 = 0
(2) 135 351 xyz ==
(3) 211 211 xyz −++ ==
(4) 211 211 xyz ==
Numerical Value Questions
7. A parallelopiped is formed by the planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of diagonal of the parallelopiped is ______.
8. If the distance betwen the plane Ax–2y+z = d and the plane containing the lines
123 234 xyz == and 234 345 xyz == is 6 then |d| is ____.
9. Let the equation of the plane containing the line x–y–z–4 = 0 = x +y+2z–4 and is parallel to the line of intersection of the planes 2x+3 y+z = 1 and x+3y+2z = 2 be x+Ay+Bz+C = 0. Compute the value of |A+B+C| is ______.
10. Find the shortest distance of plane parallel to z–axis and containing line x+y+2z–3 = 0 = 2x+3y+4z–4 from z–axis is ______.
11. The value of k if the lines
THERY-BASED QUESTIONS
Very Short Answer Questions
1. If ,, αβγ are the angles made by the line L with coordinate axes in the positive direction then what is the value of cos2cos2cos2 α+β+γ
2. If ,, lmn are direction cosines of a line L , then what is the maximum value of lmn
3. If ,, abc are direction ratios of a line L then direction cosines of that line are ,, abc kkk ±±± then find k
4. If 111 ,, abc and 222 ,, abc are direction ratios of two lines, then the condition for those two lines are to be perpendicular.
123 322 xyz k ++− == and 156 317 xyz k −++ == may be perpendicular ______.
12. The point on the line 235 122 xyz −++ == at a distance of 6 from the point (2, –3, –5) is ( α , β , γ ) then the non–zero numerical value of βγ αβ + = + = _______.
13. A plane passes through (1, –2, 1) and is perpendicular to two planes 2x–2y+z = 0 and x–y+2z = 4 The distance of the plane from the point (1, 2, 2) is d, then 2 d = = ______.
14. A parallelopiped is formed by the planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinate planes. The length of diagonal of the parallelopiped is______.
5. If ,, 34 ππθ are the angles made by a line with coordinate axes in the positive direction, then find cos θ
6. The direction cosines of a line perpendicular to the plane 0 axbyczd+++= are ,, abc kkk ±±± the write the value of k.
7. What is the perpendicular distance from origin to the plane 0 axbyczd+++= .
8. What is the equation of a plane passing through () ,, abc and having normal whose direction ratios are () ,, abc .
9. What is the equation of the plane passing through the points ()() 3,0,0,0,4,0 and () 0,0,5
10. What is the distance of a point () 1,3,10 to the plane 3 xyz−+= along the line 1 22 3412 y xz + ==
11. What is the vector equation of the line passing through two points () 3,4,7 A and () 1,1,6 B
12. What are the direction cosines of a line parallel to the line 2 213 13 3 y xz + +− ==
13. Find the angle between the lines 23xyz ==− and 64 xyz =−=−
Statement Typed Questions
Each question has two statements: statement I (S-I) and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct
(2) if both statement I and statement II are incorrect
(3) if statement I is correct, but statement II is incorrect
(4) if statement I is incorrect, but statement II is correct
14. S - I : 111 ,, 2 22 are not the direction cosines of any line
S - II : If ,, lmn are direction cosines of a line then 222 1 lmn++=
15. S - I : The condition for two lines having direction ratios 111 ,, abc and 222 ,, abc are to be parallel is 111 222 abc abc ==
S - II : Direction ratios of a line perpendicular to the lines having direction ratios 111 ,, abc and 222 ,, abc are proportional to 122112211221 ,, bcbccacaabab
16. S - I : The direction cosines of a line are unique set of ordered triad.
S - II : direction ratios of a line are infinite.
17. S - I : Direction ratios of line of support of vector 123 ˆ ˆ ˆ bbibjbk =++ are proportional to 123 ,, bbb
S - II : A vector along a line having direction ratios ,, abc is ˆ ˆ ˆ aibjck ++
18. S - I : If 3 OP = having direction ratios proportional to 1,2,2 , then the coordinates of a point P is () 1,2,2 or () 1,2,2
S - II : If direction cosines of a line which is passing through the origin are ,, lmn then the coordinates of a point at a distance of k from origin is () ,, lkmknk±±±
19. S - I : The angle between two planes 25xyz++= and 313 xyz−+= is 3 π
S - II : The angle between the vectors ˆ ˆ ˆ 2 ijk ++ and ˆ ˆ ˆ 313 ijk−+= is 3 π
20. S - I : The nearest point to () 2,3,1 on the plane 210 xyz−++= is () 1,3,0
S - II : The foot of the perpendicular to the point () 2,3,1 on the plane 210 xyz−++= is () 1,3,0
21. S - I : The point () 3,1,6 A is the mirror image of the point () 1,3,4 B in the plane 5 xyz−+=
S - II : The plane 5 xyz−+= bisects the line segment joining the points () 3,1,6 A and () 1,3,4 B
22. S - I : The equation of plane which is at a distance of 33 units from the origin and whose normal makes equal angles with coordinate axes is 9 xyz++=
S - II : The equation of plane in normal form is rnp ⋅= where p is perpendicular distance from origin and n is unit normal vector to the plane.
23. S - I : The line 1 2 123 y xz + == is parallel to the plane 0 xyz+−=
S - II : The condition for the line 111 xxyyzz lmn == is to be parallel to the plane 0 axbyczd+++= is 0 albmcn++=
24. S - I : The vector equation of the line 4 56 272 y xz + == is
546372 rijktijk =−+++−
S - II : The equation of a line passing through the point ()Aa and parallel to b is ratb =+
25. S - I : The equation of plane which is perpendicular bisector of the line
segment joining the points () 2,3,4 and () 6,7,8 is 15 xyz++=
S - II: The equation of plane which is perpendicular bisector of the line segment joining the points () 111 ,, Axyz and () 222 ,, Bxyz is in the form of ()()() 212121 xxxyyyzzzk −+−+−= , this is passing through the midpoint of AB
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A)
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A)
(3) if (A) is true but (R) is false
(4) if both (A) and (R) are false
26. (A) : If a line makes angles ,, αβγ with axes in the positive direction then 222 sinsinsin2 α+β+γ=
(R) : If ,, lmn are direction cosines of a line then 222 1 lmn++=
27. (A) : There are four lines which makes equal angles with coordinate axes in three–dimensional geometry.
(R) : The direction cosines of a line which makes equal angles with axes is 111 ,, 333 ±±±
28. (A) : If (a, b, c) is a point on the line which is passing through the origin then the direction ratios of that line are proportional with (a, b, c)
(R) : If (l, m, n) are direction cosines of a line which is passing through the origin, then coordinates of points which are at 1 unit distance from the origin are (l, m, n) and (–l, –m, –n)
29. (A) : If a line makes 3 π with each x-axis and y-axis then the angle made by the same line with z-axis is 4 π .
(R) : Sum of the angles made by the line with axes in the positive direction is p .
30. (A) : The angle between the diagonals of cube is 1 1 cos 3
(R) : If a line makes angles ,,, αβγδ with the diagonal of a cube then 2222 4 coscoscoscos 3 α+β+γ+δ= .
31. (A) : Two planes 2240 xyz+−+= and 3290 xykz+++= are perpendicular to each other then k = –4
(R) : If the planes 1111 0 axbyczd+++= and 2222 0 axbyczd+++= are perpendicular, then 111 222 abc abc ==
32. (A) : Equation of a yz-plane is ˆ 0 rj⋅=
(R) : The equation of plane whose unit normal vector is n is 0 rn⋅=
33. (A) : If the vector normal to the plane is equally inclined to the coordinate axes then the equation of that plane will be c + y + z = k
(R) : For the plane 0 axbyczd+++= , the normal vector is ˆ ˆ ˆ aibjck ++
34. (A) : The equation of the plane passing through the points
()() 1,0,2,3,1,1AB and () 1,2,1 C is 32411 xyz++=
(R) : The equation of the plane passing through the points () 111 ,, Axyz , () 222 ,, Bxyz and () 333 ,, Cxyz is 111 212121 313131 0 xxyyzz xxyyzz xxyyzz −−−=
35. (A) : The plane x +2y – 2 = 0 contains the line 23035 xyzxyz −+−==++= and is perpendicular to the plane 230 xyz−+−=
(R) : Equation of a plane containing the line 12 0 π==π is 12 0 π+λπ=
36. (A) : The line 1 2 123 y xz + == is perpendicular to the plane 36980 xyz++−=
(R) : The condition for the line 111 xxyyzz lmn == is to be perpendicular to the plane ax + by + cz + d = 0 is al + bm + cn = 0
JEE ADVANCE LEVEL
1. If a line makes angles α1, α2, α3, α4 with diagonals of a cube then
(1) 4 1 4 cos2 3 i i α = ∑=−
(2) 4 1 4 sin2 3 i i α = ∑=−
(3) 4 2 1 sin 3 8 i i α = ∑=
(4) 4 2 1 sin 3 8 i i α = ∑=
2. If the line ()
232 ˆ ˆˆˆˆ rijkijk λ =−++++ makes angles α, β,γ with xy, yz, zx plane, respectively, then which one of the following not possible?
(1) sin2α+sin2β +sin2γ = 2 and cos2α+cos2β +cos2γ = 1
(2) tan2α+tan2β +tan2γ = 7 and cot2α+cot2β +cot2 γ = 5 3
(3) sin2α+sin2β +sin2γ = 1 and cos2α+cos2β +cos2 γ = 2
(3) () 62,6,6
(4) () 62,6,6
5. The direction cosines of a line equally inclined with the coordinate axes are (1) 111 ,, 333 (2) 111 ,, 333 (3) 111 ,, 333 (4) 111 ,, 333
6. If the direction cosines l, m, n of a line are related by the equations l+m+n = 0, 2mn+2 ml–nl = 0 Then the ordered triplet (l, m, n) is
(1) 112 ,, 666
(2) 112 ,, 666
(4) 222 222 14 secsecsecandcoscoscos10 3 ececec α+β+γ=α+β+γ=
222 222 14 secsecsecandcoscoscos10 3 ececec α+β+γ=α+β+γ=
3. If O is the origin and the line OP of length r makes an angle α with x-axis and lies in the xy-plane then the coordinates of P are not equal to (1) (rcosα, 0, rsinα) (2) (rcosα, rsinα, 0) (3) (0, 0, rcosα) (4) (rsinα, rcosα, 0)
4. If a point P is such that OP is inclined to positive x-axis at 45° and to positive y–axis at 60° and OP = 12 then coordinates of P = _____.
(1) () 62,6,6 (2) () 62,6,6
(3) 211 ,, 666 (4) 211 ,, 666
7. The direction cosines of two lines are connected by relations l+m+n = 0 and 4l is the harmonic mean between m and n, then
(1) 111 222 3 2 lmn lmn ++=−
(2) 121212 1 2 llmmnn++=−
(3) 111222 6 9 lmnlmn+=−
(4) ()()() 121212 6 18 llmmnn +++++=
8. The plane passing through the origin and containing the lines whose direction cosines are proportional to (1, –2, 2) and (2, 3, –1) passes through the point (1) (1, –2, 2) (2) (2, 3, –1) (3) (3, 1, 1) (4) (4, 0, 7)
9. The plane is perpendicular to the line 2 1 y xz rr == passes through the origin and the point (–4, 3, 1), if r is equal to (1) 1 (2) 5 (3) –4 (4) 3
10. Consider the planes 3x – 6y + 2z + 5 = 0 and 4x – 12y + 3z = 3. The plane 67x – 162y + 47z + 44 = 0 bisects the angle between the given planes which (1) contains origin (2) is acute (3) is obtuse (4) right angle
11. Equation of plane through A(1, 0, 0), B(0, 1, 0) and making an angle 4 π with the plane x+y = 3 can be (1) x–y+2z = 1 (2) x+y+ 2 z = 1
(3) x+y– 2 z = 1 (4) x–y–2z = 1
12. The planes ax+4y+z = 0, 2y+3z–1 = 0, 3x–bz+2 = 0.
(1) Will meet at a point, if ab ≠ 15
(2) Will meet on a line, if ab = 15, a = 3
(3) Will have no common point, if ab = 15, a ≠ 3
(4) Will have no common point, if ab = 15, b ≠ 5
13. Consider the two planes 1π : 2x–y+2z+3 = 0 ; 2π : 3x–2y+6z+8 = 0
(1) The obtuse angular bisector is 5 x–y–4z–3 = 0
(2) The acute angular bisector is 5x–y–4z–3 =0
(3) The bisector of the angle containing the origin is 5x–y–4z–3 = 0
(4) The Drs of line of intersection of planes (2, –6, 1)
14. If (α, α2 , α) and (3, 2, 1) lie on same side of the plane π : x + y –4 z +2 = 0 then the interval(s) to which α can belong
(1) [–10, 0] (2) [2020, 2021] (3) [–3, 3] (4) (27, 214)
15. Equation of a plane through the line 123 234 xyz == and parallel to a coordinate axes is (1) 4y–3z+1 = 0
(2) 2x–z+1 = 0
(3) 3x–2y+1 = 0 (4) 2x+3y+1 = 0
16. The equation of the line x+y+z–1 = 0, 4x+y–2z+2 = 0 written in the symmetrical form is
(1) 120 121 xyz +−− ==
(2) 1 121 xyz ==
(3) 11 1 22 121 xz y +− ==
(4) 122 212 xyz −+− ==
17. For the lines 231 y xz == and 1 22 352 y xz−+ ==
(1) Dr’s of line of shortest distance are 111 ,, 333
(2) Line of shortest distance pass through the point 9232 21,, 33
(3) Equation of line of shortest distance is 3(x – 21) = 3y + 92 = 3z – 32
(4) Dr's of line of shortest distance are (2, 2, 2)
18. If y = 2x, z = 5, and y = –2x, z = –5 are two lines, then
(1) The acute angle between the lines is greater than 4 π
(2) Both lines are parallel to z-axis
(3) The lines are intersecting lines
(4) Locus of point equidistant from the two lines is 2xy+25z = 0
19. The plane x – 2y + 7z + 21 = 0
(1) contains the line 132 321 xyz +−+ ==
(2) contains the point (0, 7, –1)
22. The lines 11 112 xyz −+ == and 11 22 xyz λ == are
(1) Parallel, if λ = 4
(2) Perpendicular, if λ = –2
(3) Coplanar, if λ = 4
(4) Skew lines, if λ = 5
23. Consider the lines 12 212 : ,:43 175 xyz L Lxyz −−+ ==−=+=− 12 212 : ,:43 175 xyz L Lxyz −−+ ==−=+=−
(3) is perpendicular to the line 127 xyz ==
(4) is parallel to the plane x – 2y + 7z = 0
20. Consider the lines x = y = z and the line
2x + y + z –1 = 0 = 3x + y + 2z – 2 is
(1) The shortest distance between the two lines is 1 2
(2) The shortest distance between the two lines is 2
(3) Plane containing 2nd line parallel to 1st line is y – z + 1 = 0
(4) The shortest distance between the two lines 3 2
21. If lines 1 3 23 y xza p + −+ == and 4 25 242 y xz++ == are perpendicular coplanar lines, then (1) p = 4
(2) 2 7 p =− (3) 4 7 a = (4) 30 7 a =−
Then which of the following is/are correct
(1) Point of intersection of L1 and L2 is (1. –6, 3)
(2) Equation of plane containing L1 and L2 is x+2y+3z+2 = 0
(3) Acute angle between L1 and L2 is 1 13 cot 15
(4) Equation of plane containing L1 and L2 is x+2y+2z+3 = 0
24. Projection of the line 113 214 xyz+++ == on the plane x+2y+z = 6 has the possible equation is
(1) x+2y+z–6 = 0 = 9x–2y–5z–8
(2) x+2y+z–6 = 0 = 9x–2y+5z–8
(3) 131 4710 xyz −−+ ==
(4) 327 4410 xyz +−− ==
25. If a line with dr's (1. –5, –2) meets the lines x = y+5 = z+1 and x+5 = 3y = 2z at A and B respectively, then
(1) A(2, –3, 1) (2) B(1, 2, 3)
(3) AB = 30 (4) A(0, –3, 1)
26. The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured along the line parallel to 236 xyz == is _____.
27. Let P is a point on the plane ax+by+cz = d. A point Q(α, β, γ) is taken on the line OP such that OP.OQ = d2, then () 222 2 dabcαβγ αβγ ++ ++ is equal to (where O is origin)
28. If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (α, 7, 1) is 5717 ,, 333 , then α = _____.
29. A variable plane cutting coordinate axes in A,B,C is at a constant distance 1 from the origin. Then the locus of centroid of the triangle ABC is x–2+y–2+z–2 = k then k = _____.
30. Image of the point (p,q,r) with respect to the plane 2x+y+z = 6 is (p', q', r' ) and if 5p+q+r = 15 then the value of p' + q' + r' = _____.
31. If the lengths of the sides of a rectangular parallelopiped are 3, 2, 1 then the angle between two diagonals out of four diagonals is 1 6 cos k , then k = _____.
32. If A = (3, 1, –2), B =(–1, 0, 1) and l, m are the projections of AB on the Y -axis, ZXplane, respectively, then 3l2 –m+2 = _____.
33. If the dc's of two parallel lines are given by 2 l+3m+kn = 0 and l2 –m2+5n2 = 0, then the value of 2 5 k = _____.
34. A = (1, 2, 3), B = (4, 5, 7), C = (–4, 3, –6), D = (2, k, 2) are four points. If the lines AB and CD are parallel, then k = _____.
35. If a line makes angles α , β , γ , δ with the four diagonals of a cube, then sin2α + sin2β – cos2γ –cos2δ = P Q ⇒ P+Q = ______. (P and Q are coprime)
36. If the length of the perpendicular drawn from the point P(α, 4, 2) (α>0) on the line
131
231 xyz +−− == is 26 units and Q(a,
b, c) is the image of the point P in this line then α +a+b+c is equal to _____.
37. Let PM be the perpendicular from the point P(1, 2, 3) to xy-plane. If OP makes an angle θ with the positive direction of the z-axis and OM makes an angle φ with the positive direction of x-axis, where O is the origin, then 35 tan θ +tan φ is ______.
38. If the line 1 21 321 y xz + == intersects the curve xy=c2 = z = 0 then c2 is equal to ______.
(Q. : 39-40)
Consider the line 1 121 312 xyz L +++ === 2 223 123 xyz L −+− ===
39. The shortest distance between L1 and L2 is 53 λ then λ is ____
40. The distance of the point (1, 1, 1) from the plane passing through the point (–1, –2, –1) and whose normal is perpendicular to both the lines L1 and L2 is 75 λ then λ is ____
(Q. : 41-42)
If A(5, 4, 6), B = (1, –1, 3) and C(4, 3, 2) are vertical of ∆ ABC and the internal bisector of angle A meets side BC in D then
41. 170 AD = ______.
42. 1 51 ( The area of ∆ ABC ) = ______.
(Q. : 43-44)
Given four points A(2, 1, 0), B(1, 0, 1), C(3, 0, 1) and D (0, 0, 2) point D lies on a line L orthogonal to the plane determined by the points A, B, C
43. The equation of plane ABC is (1) x+y+z–3 = 0 (2) y+z–1 = 0 (3) x+z–1 = 0 (4) 2y+z–1 = 0
44. The equation of the line L is (1) r = 2k+λ(i+k) (2) r = 2k+ λ(2j+k) (3) r = 2k+ λ(j+k) (4) r = –k
(Q. :45-46)
Dr's of a normal to the plane ax+by+cz+d = 0 are (a, b, c)
45. The plane passing through the points (0, –1, 2) and (–1, 2, 1) and parallel to the line passing through (5, 1, –7) and (1, –1, –1) also passes through the point (1) (1, –2, 1) (2) (0, 5, –2) (3) (–2, 5, 0) (4) (2, 0, 1)
46. The distance of the point (7, –3, –4) from the plane passing through the points (2, –3, 1), (–1, 1, –2) and (3, –4, 2) is (1) 4 (2) 5 (3) 52 (4) 42
(Q. : 47-48)
Consider the lines 1 121 : 312 xyz L +++ ==
2 223 ; : 123 xyz L −+− ==
47. The unit vector perpendicualr to both L1 and L2 is (1) 77 ˆ 9 ˆˆ 9 ijk−++ (2) 75 ˆ 5 ˆˆ
(3) 75
3
48. The shortest distance between L1 and L2 is (1) 0 (2) 17 3 (3) 41 53 (4) 17 53
(Q. :49-50)
Let L 1 and L 2 be the lines x +2 y – z –3 = 0 = 3 x–y+2z–1 and 2x–2y+3z–2 = 0 =
49. Square of the distance of the origin from the point of intersection of L1 and L2 is _____.
50. The distance of the origin from the plane through the lines is 1 ab units, then a+b is equal to ______.
51. Match the following List - I with List - II
List - I List - II
(A) A directed line makes angles 60° and 45° with the x–axis and y–axis respectively the angle it make with z–axis is (p) 1 9 cos
(B) The acute angle between the lines whose direction cosines are given by l+m+n = 0, l2+m2 –n2 = 0 is (q) 1 55 cos
(C) If A,B,C are the feet of the perpendiculars from (3, 4, 5) to coordinate axes then angle between AB, AC is (r) 3 π
(D) If A(4, 3, 5) B(0, 6, 0) C(–8, 1, 4) are the consecutive vertices of a parallelogram ABCD then the angle between AC and BD is (s) 2 π
CHAPTER 13: 3D Geometry
A B C D
(1) r r p q
(2) p q r s
(3) r s p q
(4) q r s p
52. Match the following List - I with List - II
List - I List - II
(A) The angle between diagonal of the cube and a diagonal of a face skew to it is (p) 1 cot2
(B) The angle between the diagonlas of two faces of the cube through the same vertex (q) 1 1 cot 22
(C) The angle between a diagonal of a cube and a diagonal of a face intersecting it is (r) () 1 cot0
(D) The angle between a diagonal of a cube and a diagonal of a face intersecting it is (s) 1 1 cot 3
A B C D
(1) r s p q
(2) p q r s
(3) r s p q
(4) q r s p
53. Match the following List - I with List - II
List - I List - II
(A) L1: x = 1+t, y = t, z = 2–5t
L2: r = (2, 1, –3) + λ(2, 2–10) (p) Non coplanar lines
(B) 1 2 132 : 221 262 : 113 xyz L xyz L == −−+ == (q) Line lie in a unique plane
(C) L1: x = –6t, y = 1+9t, z = –3t
L2: x = 1+2s, y = 4–3s, z = s (r) Infinite planes containing both the lines
(D) 1 2 12 :; 123 321 : 432 xyz L xyz L == == (s) Lines do not intersect
A B C D
(1) r q q,s p,s
(2) p q r s
(3) r s p q
(4) q r s p
54. Match the following List-I with List-II List-I List-II
(A) The coordinates of a point on the line x = 4y+5, z = 3 y–6 at a distance 3 from the point (5, 3, –6)
(B) The plane containing the lines 235 357 xyz −++ == 235 357 xyz −++ == and parallel to ˆ 7 ˆ 4 ˆ ijk ++ has the point
(C) A line passes through two points A line passes through two points A(2, −3, −1) and B(8, −1, 2). The coordinates of a point on this line nearer to the origin and at a distance of 14 units from A is/ are
(D) The coordinates of the foot of the perpendicular from the point (3, –1, 11) on the line 23 234 xyz == is/are
(p) (–1, –2, 0)
(q) (5, 0, –6)
(r) (2, 5, 7)
(s) (14, 1, 5)
A B C D
(1) s r q p
(2) r q p s
(3) q p r s (4) p s r q
55. Match the following List - I with List - II List - I List - II
(A) The shortest distance between the two straight lines 31 6 42 237 −+ ==− xy z and 432176 576 xyz −+−
9 2
(B) Two lines 111 234 xyz−+− == and 3 12 xyk z == intersect at a point then k = (q) 0
(C) The sphere x2+y2+z2 = 25 intersects the plane 3x–4z+5 = 0 in a circle then its radius is (r) 2
(D) The line x = y = z meets the plane x+y+z = 1 at P and the sphere x2+y2+z2 = 1 at the points R and S then PR + PS = (s) 26 A B C D
(1) q p s r
(2) p q s r
(3) s q p q (4) p s r q
56. Match the following List - I with List - II
List - I List - II
(A) Angle between any two solid diagonal (p) 1 2 cos 6
(B) Angle between a solid diagonal and a plane (q) 1 1 cos 2
(C) Angle between plane diagonals of adjacent faces (r) 1 1 cos 3
(D) If a line makes angle 4 π and 3 π with positive x and y axis then the angle which it makes with positive z-axis (s) 1 2 A B C D
(1) r p q q
(2) q p r s
(3) s q r p
(4) s r q p
57. Match the following List - I with List - II
List - I List - II
(A) The coordinates of a point on the line x = 4y+5, z = 3 y–6 at a distance 3 from the point (5, 3, –6) is/are (p) (–1, –2, 0)
(B) The plane containing the lines
(q) (5, 0, –6)
235 357 xyz −++ ==
235 357 xyz −++ == and parallel to ˆ 7 ˆ 4 ˆ ijk ++ has the point
(C) A line passes through two points A(2, –3, –1) and B(8, –1, 2). The coordinates of a point on this line nearer to the origin and at a distance of 14 units form A is /are
(D) The coordinates of the foot of the perpendicular from the point (3, –1, 11) on the line 23 234 xyz == is/are
(r) (2, 5, 7)
(s) (14, 1, 5) A B C D
(1) q p s r
(2) s q r p
(3) p q r s
(4) r q s p
BRAIN TEASERS
1. A mirror and source of light are kept at the origin and at a point on the positive x-axis respectively. A ray of light from the sources strikes the mirror and is reflected. If (1, –1, 1) are the dr's of a normal to the plane, then the dc's of the reflected ray are:
(1) 122 ,, 333
(2) 122 ,, 333
(3) 122 ,, 333
(4) 111 ,, 333
2. A hall has a square floor of dimension 10m × 10m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is 1 1 cos, 5 then the height of the hall (in meters) is:
(1) 52 (2) 53
(3) 5 (4) 210
3. Let PN be the perpendicular from the point P(1, 2, 3) to xy-plane. If OP makes an angle α with positive direction of the z-axis and ON makes an angle β with the direction of x-axis, where O is the origin α and β acute angles), then
(1) 2 sinsin 14 αβ=
(2) 2 coscos 14 αβ=
(3) 5 tan 3 α= (4) tan β = 2
4. Consider a plane P passing through, A(λ, 3, μ), B(–1, 3, 2), and C(7, 5, 10) and a straight line L with positive direction cosines passing through A, bisecting BC and makes equal angles with the coordinate axes. Let L1 be
a line, parallel to L and passing through origin. Then which of the following is/are true?
(1) The value of ( λ +μ) is 5.
(2) Equation of the straight line L 1 is 111 111 xyz ==
(3) Equation of the plane perpendicular to the plane P and containing the line L is x–2y+z = 0
(4) Area of the triangle ABC is 32 square units
5. A ray is sent along the line 021 220 xyz == and is reflected by the plane x = 0 at a point
A. The reflected ray is again reflected by the plane x+2y = 0 at point B. The initial ray and final reflected ray meets at point J then which of the following is/are true
(1) The coordinates of the point B is (4, –2, 1)
(2) The coordinates of the point J is
(3) The centroid of ∆ ABJ is (0, 0, 0)
(4) The coordinates of J is (2, –1, 1)
6. Let L1 and L2 be the following straight lines
1 11 : 113 xyz L == and
2 11 : 311 xyz L == suppose the straight
line 1 : 2 xyz L lm αγ == lies in the plane
containing L 1 and L 2 and passes through the point of intersection of L1 and L2 If the line L bisect the acute angle between the lines L1 and L2 then which of the following statements is/are TRUE
(1) α – γ=3
(2) l+m = 2
(3) α – γ=1
(4) l+m = 0
FLASHBACK (Previous JEE Questions)
1. The distance of the point (7, –2, 11) from the line 648 103 xyz == along the line 515 236 xyz == , is:
(27th Jan 2024 Shift 1)
(1) 12 (2) 14 (3) 18 (4) 21
2. If the shortest distance between the lines 41 123 xyz −+ == and 12 245 xyz λ −+− == is 6 5 , then the sum of all possible values of λ is: (27th Jan 2024 Shift 1)
(1) 5 (2) 8 (3) 7 (4) 10
3. Let PQR be a triangle with R(–1, 4, 2). Suppose M(2, 1, 2) is the mid point of PQ. The distance of the centroid of ∆ PQR from the point of intersection of the line 23 021 xyz−+ == and 131 131 xyz −++ ==
(29th Jan 2024 Shift 1)
(1) 69 (2) 9 (3) 69 (4) 99
4. Let P(3, 2, 3), Q(4, 6, 2) and R(7, 3, 2) be the vertices of ∆ PQR. Then, the angle ∠ QPR is (29th Jan 2024 Shift 2) (1) 6 π (2) 1 7 cos 18
(3) 1 cos 18 1 (4) 3 π
5. Let ( α , β , γ ) be the foot of perpendicular from the point (1, 2, 3) on the line 314 523 xyz +−+ == then 19( α + β + γ ) is equal to (30th Jan 2024 Shift 1) (1) 102 (2) 101 (3) 99 (4) 100
6. Let (α, β, γ) be mirror image of the point (2, 3, 5) in the line 123 234 xyz . Then 2 α +3 β +4 γ is equal to (31st Jan 2024 Shift 2) (1) 32 (2) 33 (3) 31 (4) 34
7. The shortest distance between lines L1 and L2 where 1 114 : 232 xyz L −++ == and L2 is the line passing through the points A(–4, 4, 3), B(–1, 6, 3) and perpendicular to the line 31 231 xyz == is (31st Jan 2024 Shift 2)
(1) 121 221 (2) 24 117 (3) 141 221 (4) 42 117
8. If the shortest distance between the lines 21 211 xyz λ == and 312 121 xyz == is 1, then the sum of all possible values of λ is (01st Feb 2024 Shift 1) (1) 0 (2) 23 (3) 33 (4) 23
9. Let P and Q be the points on the line 341 822 xyz +−+ == which are at a distance of 6 units from the point R (1, 2, 3). If the centroid of the triangle PQR is ( α , β , γ ) , then the value of α 2+ β 2+ γ 2= (01st Feb 2024 Shift 2) (1) 26 (2) 36 (3) 18 (4) 24
10. If the mirror image of the point P(3, 4, 9) in the line 112 321 xyz−+− == is ( α , β , γ ), then 14( α + β + γ ) is (20th Apr 2023 Shift 2) (1) 102 (2) 138 (3) 108 (4) 132
(27th Jan 2024 Shift 2)
11. The lines 27 2216 xyz == and 322 431 xyz +++ == intersect at the point P. If the distance of P from the line 111 231 xyz+−− == is l , then 14 l 2 is equal to _____.
12. A line with direction ratios 2, 1, 2 meets the lines x = y + 2 = z and x+2 = 2y = 2z respectively at the point P and Q . if the length of the perpendicular from the point (1, 2, 12) to the line PQ is l, then l2 is _____.
(29th Jan 2024 Shift 1)
13. If d1 is the shortest distance between the lines x+1 = 2y = –12z, x = y+2 = 6z–6 and d2 is the shortest distance between the lines 184126 , 275213 xyzxyz −+−−−− ==== , then the value of 1 2 323d d is _____.
(30th Jan 2024 Shift 1)
14. Let a line passing through the point (–1, 2, 3) intersect the lines 1 121 : 322 xyz L −−+ == at M(α, β, γ) and 2 221 : 324 xyz L +−− == at N(a, b, c). Then the value of 2 2 () () abc αβγ ++ ++ equals (30th Jan 2024 Shift 2)
15. A line passes through A(4, –6, –2) and B(16, –2,4). The point P(a, b, c) where a, b, c are non-negative integers, on the line AB lies at a distance of 21 units, from the point A The distance between the points P(a, b, c) and Q(4, –12, 3) is equal to ___
(31st Jan 2024 Shift 2)
16. The distance of the point (–1, 9, –16) from the plane 2x + 3y – z = 5 measured parallel to the line 423 3412 xyz +−− == is
(24th Jan 2023 Shift 1)
(1) 202 (2) 31 (3) 132 (4) 26
17. The distance of the point (7, –3, –4) from the plane passing through the points (2, –3, 1), (–1, 1, –2) and (3, –4, 2) is (24th Jan 2023 Shift 1)
(1) 52 (2) 42 (3) 4 (4) 5
18. If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes x+2y+z = 0 and 3y–z = 3 is ( α , β , γ ), then α + β + γ is equal to (24th Jan 2023 Shift 2) (1) –1 (2) 1 (3) 3 (4) 5
19. Let the plane containing the line of intersection of the planes P1: x+(λ+4)y+z = 1 and P2:2x+y +z = 2 pass through the points (0, 1, 0) and (1, 0, 1). Then the distance of the point (2 λ , λ ,– λ ) from the plane P2 is (24th Jan 2023 Shift 2)
(1) 56 (2) 46 (3) 36 (4) 26
20. Consider the lines L1 and L2 given by
1 132 : 212 xyz L ==
2 223 : 123 xyz L ==
A line L 3 having direction ratios (1, –1, –2), intersects L 1 and L 2 at the points P and Q respectively. Then the length of line segment PQ is
(25th Jan 2023 Shift 1)
(1) 32 (2) 4
(3) 26 (4) 43
21. The distance of the point (4, 6, -2) from the line passing through the point (-3, 2, 3) and parallel to a line with direction ratios (3, 3, -1) is equal to:
(25th Jan 2023 Shift 1)
(1) 23 (2) 14
(3) 3 (4) 6
22. The shortest distance between the lines x+1 = 2y = –12z and x = y+2 = 6z–6 is
(25th Jan 2023 Shift 2)
(1) 5 2 (2) 3 (3) 3 2 (4) 2
23. The foot of perpendicular of the point (2, 0, 5) on the line 111 251 xyz+−+ == is ( α , β , γ ) Then which of the following is NOT correct?
27. Let a unit vector OP makes angles α , β , γ with the positive directions of the coordinate axes OX, OY, OZ respectively, where 0, 2 βπ ∈ . If OP is perpendicular to the plane though the points (1, 2, 3),(2, 3, 4) and (1, 5, 7), then which one of the following is true? (30th Jan 2023 Shift 1)
(25th Jan 2023 Shift 2)
(1) 4 15 αβ γ = (2) 5 β γ =−
(3) 5 8 γ α =− (4) 8 α β =−
24. If the lines 123 121 xyz −−+ == and 23 231 xayz −+− == intersect at the point P, then the distance of the point P from the plane z = a is (29th Jan 2023 Shift 2)
(1) 28 (2) 16 (3) 22 (4) 10
25. The shortest distance between the lines 184 275 xyz −+− == and 126 213 xyz == is
(29th Jan 2023 Shift 2)
(1) 33 (2) 43 (3) 23 (4) 53
26. The plane 2 x – y + z = 4 intersects the line segment joining the points A(a, –2, 4) and B(2, b, –3) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is 5 . If ab <0 and P is the point (a–b, b, 2b–a) then CP2 is equal to (29th Jan 2023 Shift 2) (1) 17 3 (2) 97 3 (3) 73 3 (4) 16 3
(1) 0, 2 π α ∈
and , 2 π γπ ∈
(2) 0, 2 π α ∈ and 0, 2 π
(3) , 2 π απ
(4) , 2
and 0, 2
and , 2
28. The line l1 passes thorugh the point (2, 6, 2) and is perpendicular to plane 2x+y–2z = 10. Then the shortest distance between the line l1 and the line 14 232 xyz ++ == is: (30th Jan 2023 Shift 1)
(1) 7 (2) 13 3 (3) 9 (4) 19 3
29. A vector v in the first octant is inclined to the x–axis at 60°, to the y–axis at 45° and to the z–axis at an acute angle. If a plane passing through the points () 2,1,1 and (a, b, c), is normal to v , then (30th Jan 2023 Shift 2)
(1) 2 a–b+c = 1 (2) 2 a+b+c = 1 (3) a+b+ 2 c = 1 (4) a+ 2 b+c = 1
30. If a plane passes through the points (–1, k, 0), (2, k, –1),(1, 1, 2) and is parallel to the line 1211 121 xyz −++ == , then the value of () () 2 1 12 k kk + is (30th Jan 2023 Shift 2)
(1) 6 13 (2) 17 5 (3) 5 17 (4) 13 6
31. Let the shortest distance between the lines 5 :, 201 xyz L λλ−−+ == , λ≥ 0 and L1: x+1 =
y–1 = 4–z be 2 6 . If (α, β, γ) lies on L, then which of the following is NOT possible? (31st Jan 2023 Shift 1)
(1) α+2 γ = 24 (2) 2α– γ = 9 (3) α–2 γ = 19 (4) 2α+ γ = 7
32. The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points A, B,C is (2, a, 4), a ∈ N. If the volume of the tetrahedron OABC is 144 unit3, then which of the following points is NOT on P? (31st Jan 2023 Shift 2) (1) (0, 4, 4) (2) (3, 0, 4) (3) (0, 6, 3) (4) (2, 2, 4)
33. Let P be the plane, passing through the point (1, –1, –5) and perpendicular to the line joining the points (4, 1, –3) and (2, 4, 3). Then the distance of P from the point (3, –2, 2) is
(31st Jan 2023 Shift 2)
(1) 6 (2) 4 (3) 5 (4) 7
34. Let the plane P: 8x+ α 1y+ α 2z+12 = 0 be parallel to the line
234 : 235 xyz L +−+ == . If the intercept of P on the y-axis is 1, then the distance between P and L is (31st Jan 2023 Shift 2)
(1) 2 7 (2) 6 14 (3) 7 2 (4) 14
35. The shortest distance between the lines 524 123 xyz == and 351 145 xyz ++− == is
(01st Feb 2023 Shift 1)
(1) 73 (2) 63 (3) 43 (4) 53
36. Let the image of the point P(2, –1, 3) in the plane x+2y–z = 0 be Q. Then the distance of the plane 3x+2y+z+29 = 0 from the point Q is (1st Feb 2023 Shift 1)
(1) 214 (2) 222 7
(3) 314 (4) 242 7
37. Let the plane P pass through the intersection of the planes 2x+3y–z = 2 and x+2y+3z = 6, and be perpendicular to the plane 2x+y–z+1 = 0. If d is the distance of P from the point (–7, 1, 1), then 2d is equal to: (01st Feb 2023 Shift 2)
(1) 15 23 (2) 250 83 (3) 250 82 (4) 25 83
38. The shortest distance between the lines 216 322 xyz −+− == and 618 320 xyz −−+ == is equal to ______. (24th Jan 2023 Shift 1)
39. If the shortest distance between the lines 666 234 xyz +−− == and 2626 345 xyz λ −−+ == is 6 then the square of sum of all possible values of λ is ______. (24th Jan 2023 Shift 2)
40. Let the equation of the plane passing through the line x–2y–z–5 = 0 = x+y+3z–5 and parallel to the line x + y +2 z –7 = 0 = 2 x +3 y + z –2 be ax + by + cz = 65. Then the distance of the point ( a , b , c ) from the plane 2 x +2 y – z +16 = 0 is ______. (25th Jan 2023 Shift 1)
41. If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4) and the line 112 210 xyz−+− == is α then 28 α 2 is equal to _______.
(25th Jan 2023 Shift 2)
42. Let the co-ordinates of one vertex of ∆ABC be A(0, 2, α) and the other two vertices lie on the line 14 523 xyz α +−+ == For α∈Z, if the area of ∆ ABC is 21 sq. units and the line segment BC has length 2 2 units, then α 2 is equal to ______.
(29th Jan 2023 Shift 1)
43. Let the equation of the plane P containing the line x+10 = 8 2 y = z be ax+by+3z = 2(a+b) and the distance of the plane P from the point (1, 27, 7) be c. Then a2+b2+c2 is equal to ______.
(29th Jan 2023 Shift 1)
44. If the equation of the plane passing through the point (1, 1, 2) and perpendicular to the line x–3y+2z–1 = 0 = 4x–y+z is Ax+By+Cz = 1, then 140(C–B+A) is equal to ______.
(30th Jan 2023 Shift 1)
45. Let a line L pass through the point P(2, 3, 1) and be parallel to the line x+3y–2z–2 = 0 =x–y+2z. If the distance of L from the point (5, 3, 8) is α , then 3 α 2 is equal to _____.
(30th Jan 2023 Shift 2)
46. Let the line 113 : 211 xyz L −+− == intersect the plane 2x+y+3z = 16 at the point P. Let the point Q be the foot of perpendicular from the point R(1, –1, –3) on the line L If α is the area of triangle PQR then α 2 is
equal to ______.
(31st Jan 2023 Shift 1)
47. The point of intersection C of the plane 8 x+y+2z = 0 and the line joining the points A(–3, –6, 1) and B(2, 4, –3) divides the line segment AB internally in the ratio k : 1. If a, b, c(|a|,|b|,|c| are coprime) are the direction ratios of the perpendicular from the point C on the line 142 , 123 xyz −++ == , then |a+b+c| is equal to _____.
(01st Feb 2023 Shift 2)
48. Let αx+βy+γz = 1 be the equation of a plane passing through the point (3, –2, 5) and perpendicular to the line joining the points (1, 2, 3) and (–2, 3, 5). Then the value of α βγ is equal to _____.
(1st Feb 2023 Shift 2)
JEE Advanced
49. Let Q be the cube with the set of vertices {(x1, x2, x3) ∈ 3 : x1, x2, x3 ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) in S. For lines l1 and l2, let d(l1, l 2) denote the shortest distance between them. Then the maximum value of d(l1, l2), as l1 varies over F and l2 varies over S, is
(2023 P1)
50. Let P 1 and P 2 be two planes given by P 1: 10x+15 y+12z–60 = 0, P2: –2x+5y+4z–20 = 0. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1 and P2?
(2022 P1)
(1) 111 005 xyz ==
(2) 6 523 xyz ==
(3) 4 254 xyz ==
(4) 4 123 xyz ==
(1) α – γ = 3 (2) 1+m = 2
(3) α – γ = 1 (4) 1+m = 0
53. Let α , β , γ , δ be real numbers such that α2+β2+γ2 ≠ 0 and α+γ = 1. Suppose that the point (3, 2, –1) is the mirror image of the point (1, 0, –1) with respect to the plane
αx+βy+γz = δ. Then which of the following statements is/are TRUE? (2020 P2)
(1) α + β = 2 (2) δ – γ = 3
(3) δ + β = 4 (4) α + β + γ = δ
54. Let P1: 2x+y–z = 3 and P2: x+2y+z = 2 be two planes. Then, which of the following statements(s) is (are) TRUE? (2018 P1)
(1) The line of intersection of P1 and P2 has direction ratios 1, 2, –1
(2) The line 3413 993 xyz == is perpendicular to the line of intersection of P1 and P2
(3) The acute angle between P1 and P2 is 60°
(2022 P1)
51. Let S be the reflection of a point Q with respect to the plane given by ()() ˆˆ 1 ˆ rtpitjpk =−++++ where t,p are real parameters and ˆ ˆ , ˆ , ijk are the unit vectors along the three positive coordinate axes. If the position vectors of Q and S are ˆ 10152 ˆ 0 ˆ ijk ++ and ˆˆ ˆ ijkαβγ ++ respectively, then which of the following is/are TRUE?}
(1) 3( α + β ) = –101
(2) 3( β + γ ) = –71
(3) 3( γ + α ) = –86
(4) 3( α + β + γ ) = –121
52. Let L1 and L2 be the following straight lines.
1 11 : 113 xyz L == and
2 11 : 311 xyz L == Suppose the straight line xy1Z L: m2 l αγ == lies in the plane containing L1 and L2, and passes through the point of intersection of L1 and L2. If the line L bisects the acute angle between the lines L1 and L2, then which of the following statements is/are TRUE? (2020 P1)
(4) If P 3 is the plane passing through the point (4, 2, –2) and perpendicular to the line of intersection of P1 and P2, then the distance of the point (2, 1, 1) from the plane P 3 is 2 3
55. Let P be a point in the first octant, whose image Q in the plane x+y = 3 (that is, the line segment PQ is perpendicular to the plane x+y = 3 and the mid–point of PQ lies in the plane x+y = 3) lies on the z–axis. Let the distance of P from the x–axis be 5. If R is the image of P in the xy–plane, then the length of PR is ______. (2018 P2)
56. Three lines are given by ˆ , riλλ =∈ () ˆˆ , rijµµ =+∈ and () ˆˆ , ˆ rvijkv=++∈ . Let the lines cut the plane x+y+z = 1 at the points A,B and C respectively. If the area of the triangle ABC is ∆ then the value of (6 ∆ )2 equals _____. (2019 P1)
CHAPTER TEST – JEE MAIN
Section - A
1. The angle between the lines joining the points (1, 1, 0),(–3, 3 +1, 3), and (0, –1, 0), (–1, 3 –1, λ) is 1 7 cos 16
. If λ is an integer, then λ is (1) 1 (2) 0 (3) –1 (4) 2
2. The dr's of two parallel lines are (4, –3, –1) and ( λ +μ, 1+μ, 2). Then ( λ , μ) is (1) (1, 7) (2) (–1, –7) (3) 71 , 22
(4) 95 , 22
3. If (1, –2, –2) and (0, 2, 1) are direction ratios of two lines, then the direction cosines of a line perpendicular to both the lines are (1) 112 ,, 333
(2) 212 ,, 333
(3) 212 ,, 333 (4) 213 ,, 141414
4. Find the equation of a straight line in the plane rnd ⋅= which is parallel to rab =+λ and passes through the foot of the perpendicular drawn from point ()Pa to rnd ⋅=
(where 0 nb⋅=
).
(1) 2 dan ranb n λ
(2) dan ranb
(3) 2 and ranb n
(4) and
5. If (1, 0, 0) and ( l,m,n ) are dc's of two concurrent lines and (1, –1, 0) are the dr's of acute angle bisector of the angle between them, then (l,m,n) =
(1) 122 ,, 333
(3) 122 ,, 333
(2) (0, –1, 0)
(4) 122 ,, 333
6. If the dc's (l, m,n) of two lines are connected by the relations 7l2+5m2–3n2 = 0, l–5m+3n = 0 then the dc's of the two lines are
(1) 112121 ,,;,, 666666
(2) 123112 ,,;,, 14146666
(3) 123112 ,,;,, 141414666
(4) 123112 ,,;,, 444388
7. If the direction cosines of a line L are (ab, b , b ) and the angle between L and x -axis is 3 π then a pair of possible values for a, b are
(1) 23 , 38 (2) 82 , 33
(3) 2, 5 (4) 3, 4
8. The ratio in which the plane x+2y+3z–5 = 0 divides the line segment joining the points (1, 2, 3),(–2, 3, 4), is (1) 3 : 5 (2) 7 : 5
(3) 9 : 11 externally (4) 11 : 9
9. The equation of the plane through the point (–1, 6, 2) and perpendicular to the planes x+2 y+2z–5=0 and 3x+3y+2z–8 = 0 is
(1) 2x–4y+3z+20 = 0
(2) 2x+y–3z+26 = 0
(3) 2x–4y+3z+23 = 0
(4) 2x+5y–2z+12 = 0
10. The equation of the plane which is parallel to y-axis and making intercepts of lengths 3 and 4 on x-axis and z-axis is
(1) 4x+3z = 6 (2) 4x+3z = 12
(3) 4x–3z = 12 (4) 2x–3z = 12
11. Equation of the acute angular bisector of the planes 2x–y–2z–6 = 0, 3x+2y–6z–12 = 0 is
(1) 5x–13y+4z–6 = 0
(2) 23x–y–32z–6 = 0
(3) 23x–y–32z–78 = 0
(4) 5x–13y+4z–78 = 0
12. The condition that the lines x = az+b, y = c z+d and x = a1z+b1, y = c1z+d1 to be parallel is
(1) 1 1 aa cc ==
(2) 11 1 ac ac ==
(3) aa1+cc1+1 = 0
(4) aa1 = cc1 = 1
13. The area of a triangle formed by plane 2x–3 y+4z = 12 on axes is
(1) 329 (2) 33
(3) 53 (4) 293
14. Let the line 212 352 xyz −−+ == lie in the plane x+3y–αz+β = 0. Then (α, β) equal to
(1) (6, –17) (2) (–6, 7) (3) (5, –15) (4) (–5, 5)
15. A line with direction cosines proporional to (2,1,2) meets each of the lines x = y + a = z and x + a = 2y = 2z. The coordinates of each of the points of intersection are given by
(1) (3a, 3a, 3a), (a, a, a)
(2) (2a, 3a, 3a),(2a, a, a) (3) (3a, 2a, 3a),(a, a, 2a) (4) (3a, 2a, 3a),(a, a, a)
16. The reciprocal of the distance between two points, one on each of the lines
245 325 xyz == and 123 234 xyz ==
(1) Cannot be less than 9 (2) is having minimum value 53 (3) Cannot be greater than 78 (4) Cannot be 219
17. Distance of the origin from the point of intersection of the line 23 234 xyz == and the plane 2x+y–z = 2 is (1) 120 (2) 83
(3) 219 (4) 78
18. The equation of the plane containing the line 113 214 xyz−+− == and perpendicular to the plane x+2y+z = 12 is
(1) 9x–2y+5z+4 = 0 (2) 9x–2y–5z+4 = 0
(3) 9x–2y–5z–4 = 0
(4) 9x+2y–5z–4 = 0
19. A line is drawn from the point P(1, 1, 1) and perpendicular to a line with direction ratios (1, 1, 1) to intersect the plane x+2y+3z = 4 at Q. The locus of point Q is (1) 52 121 xyz−+ ==
(2) 52 211 xyz−+ ==
(3) x = y = z
(4) 235 xyz ==
20 The lines which intersect the skew lines y = mx, z = c ; y = –mx, z = –c and the x-axis lie on the surface
(1) cz = mxy (2) xy = cmz
(3) cy = mxz (4) yz = cmx
21. The direction ratios of a line are (–2, 3, 6). If the line makes an acute angle with positive direction of x –axis then the modulus of integral value of sum of all direction cosines is ______.
22. If α, β, γ are angles, which a line makes with positive direction of the axes then |2cos ( α – β ) cos ( α + β )+2cos ( β – γ ) cos ( β + γ )+2 cos ( γ – α ) cos ( γ + α )|= ______.
23. If (l,m,n) are direction cosines of a line, what is the value of ( l+m+n ) 2 +( l+m–n)2+(m+n–l)2+(n+l–m)2 _____.
CHAPTER TEST – JEE ADVANCED
1. Let ABCD be a tetrahedron, where A = (2, 0, 0), B = (0, 4, 0), If edge CD lies on the line 123 123 xyz == Such that CD = 14 and centroid ( α , β, γ ) of tetrahedron satisfies 11 25/2 1 yz ab αβγ == , then which of the following are correct
(1) 5 2 ab+=
(2) 11 19 4 yz+=
(3) 11 1 4 yz−=
(4) a+b+y1 = 5
2. A line L passing through the point P(1, 4, 3) is perpendicular to both the lines 132 , 214 xyz −+− == and 241 . 322 xyz +−+ == If th e position
vector of point Q on L is (a1, a2, a3) such that (PQ)2 = 357, then (a1+a2+a3) can be (1) 16 (2) 15 (3) 2 (4) 1
3. The coordinates of a point on the line 11 23 xy z −+ == at a distance 414 from the point (1,–1, 0) are (1) (9, –13, 4) (2) () 8141,12141,414 +−−
24. If ( λ , μ , –6),(3, 2, –4) and (9, 8, –10) are collinear then λ+μ = _______.
25. The plane x+2y+z–3 = 0 is rotated about the line of intersection with the plane 3 x–y+z–1 = 0 through 90°, then the shortest distance from the origin, to the plane in the new position is ______.
(3) (–7, 11, –4) (4) () 8141,12141,414 −+−−
4. Let A be vector parallel to the line of intersection of planes P1 and P2. Plane P1 is parallel to the vectors 2 ˆ 3 ˆ jk + and 4 ˆ 3 ˆ jk and that P2 is parallel to ˆ ˆ jk and 3 ˆ 3 ˆ ij + then the angle between vector A and a given vector 2 ˆ 2 ˆ ˆ ijk +− is (1) 2 π (2) 4 π (3) 6 π (4) 3 4 π
5. A line with direction cosines proportional to (2, 7, –5) is drawn to intersect the lines 572 311 xyz −−+ == and 336 324 xyz == then
(1) The coordinates of point of intersection are (2, 8, –3) and (0, 1, 2)
(2) The length of the intercept on it is 78 (3) The length of the intercept on it is 68 (4) The equation of intersecting straight line be 283 275 xyz −−+ ==
6. Let the equation of a line which passes through the point (1, 1, 1) and intersects the lines 111 111 xyz == and 231 124 xyz +−+ == is L1
Which of the following statements is/are correct
(1) (4, 11, 18) is a point on the line L 1
(2) (–4, 11, 18) is point on the line L 1
(3) (–2, –9, –16) is a point on the line L 1 (4) (–2, –19, –9) is point on the line L 1
7. If the distance from point P(1, 1, 1) to the line passing through the points Q(0, 6, 8) and R (–1, 4, 7) is expressed in the form p q where p and q are coprime, then the value () () 1 2 pqpq++− = ______.
8. The distance of the point (3, 0, 5) from the line x–2y+2z–4 = 0 = x+3z–11 is _____.
9. Let A(1, 2, –1), B(2, 1, 3) and if the length of projection of segment AB on the plane x –2 y –2 z = 1 is d then the value of [ d ] is _______. (where [ ] is GIF)
10. If the lines x = 1+α, y = 3–λα, z = 1+ λα and x = 2 β , y = 1+β, z = 2–β with parameters α and β respectively, are coplanar, then the value of |λ| is ____
11. Let P(1, 2, 3) be a point in space and Q be a point on the line 131 253 xyz == . Such that PQ is parallel to 5x–4y+3z = 1. If the length of PQ is k units then k2 is _____.
12. The length of projection of the line segment joining A (1, 0, –1) and B (–1, 2, 2) on the plane π≡ x +3 y –5 z –6 = 0 is ______. (consider 5 ≅2. 23, 6 ≅2. 44, 7 ≅2. 64, 79 ≅ 8. 889)
13. The vertices B and C of a ∆ ABC lie on the line 21 304 xyz +− == such that A(1, –1, 2)
BC = 5. Then the area of the triangle ABC in sq units is ______.
14. If planes ()() ˆˆˆˆˆ , ˆ . 1,. 22rijkriajk++=++= and () 2 ˆ ˆ ˆ 3 raiajk++= intersects a line, then the number of real values of a is equal to ______.
(Q. 15-16)
The lines 321 23 xyz λ == and 232 323 xyz == lie in a same plane then
15. Equation of the plane containing both lines is
(1) x+5y–3z = 10
(2) x+6y–5z = 10
(3) x+6y+5z = 20
(4) x+5y+3z = 10
16. The value of λ is (1) 2 (2) –2 (3) 3 (4) 4
(Q. 17-18)
Consider a plane x + y –z = 1 and point A(1,2, –3). A line L has the equation x = 1 + 3 r , y = 2 – r and z = 3 + 4r
17. The coordinates of a point B of line L such that AB is parallel to the plane is
(1) (10, –1, 15)
(2) (–5, 4, –5)
(3) (4, 1, 7)
(4) (–8, 5, –9)
18. The equation of the plane containing line L and point A has the equation
(1) x–3y+5 = 0
(2) x+3y–7 = 0
(3) 3x–y–1 = 0
(4) 3x+y–5 = 0
ANSWER KEY
JEE Main Level
– I
Level – II
– III
Theory-based Questions
(26) 1
(20) 1,3 (21) 1,4 (22) 1,2,3,4(23) 1,2,3
1 (27) 2 (28) 4 (29) 9 (30) 3 (31) 7 (32) 0 (33) 5 (34) 9 (35) 5 (36) 8 (37) 7 (38) 5 (39) 17 (40) 13 (41) 0.38 (42) 1.5 (43) 2 (44) 3 (45) 3 (46) 3 (47) 2 (48) 4 (49) 33 (50) 5 (51) 1 (52) 1 (53) 1 (54) 3 (55) 1 (56) 1 (57) 1
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