13
The diagram shows a set of rectangular axes Ox, Oy and Oz, and three points ⎛ 2⎞ ⎛ 1⎞ ⎛ 1⎞ A, B and C with position vectors OA ⎜ 0⎟ , OB ⎜ 2⎟ and OC ⎜ 1⎟ . ⎝ 0⎠ ⎝ 0⎠ ⎝ 2⎠
P3 10 Exercise 10F
C
y
A x
Find the equation of the plane ABC, giving your answer in the form ax + by + cz = d. Calculate the acute angle between the planes ABC and OAB.
(i) (ii)
[Cambridge International AS & A Level Mathematics 9709, Paper 3 Q9 June 2007] 14
Two planes have equations 2x
y
3z
7 and x
2y
2z
0.
Find the acute angle between the planes. Find a vector equation for their line of intersection.
(i) (ii)
[Cambridge International AS & A Level Mathematics 9709, Paper 3 Q7 November 2008] 15
The plane p has equation 2x 3y 6z 16. The plane q is parallel to p and contains the point with position vector i + 4j + 2k. (i) (ii) (iii)
Find the equation of q, giving your answer in the form ax by cz d. Calculate the perpendicular distance between p and q. The line l is parallel to the plane p and also parallel to the plane with equation x 2y 2z 5. Given that l passes through the origin, find a vector equation for l. [Cambridge International AS & A Level Mathematics 9709, Paper 32 Q10 November 2009]
KEY POINTS 1
The position vector O P of a point P is the vector joining the origin to P.
2
The vector AB is b – a, where a and b are the position vectors of A and B.
3
The vector r often denotes the position vector of a general point.
4
The vector equation of the line through A with direction vector u is given by r a λu.
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