Chapter 3 Vectors
Norjuliyati Binti Hamzah norjuliyati@uitm.edu.my
Vectors and Their Properties Vector and Motion In one-dimensional motion, vectors were used to a limited extent For more complex motion, manipulating vectors will be more important Vector versus Scalar Review A vector quantity has both magnitude (size) and direction Examples: velocity, displacement (e.g., 10 feet north), force, magnetic field. A scalar is completely specified by only a magnitude (size) Examples: speed, distance, height.
Vectors and Their Properties
We indicate the direction of travel with a plus (+) sign for motion to the right and a negative (-) sign for motion to the left.
Vectors and Their Properties Vector Notation When handwritten, use an arrow over the letter:
When printed, will be in bold print with an arrow:
When dealing with just the magnitude of a vector in print, an italic letter will be used: A Italics will also be used to represent scalars
Vectors and Their Properties Properties of Vectors Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Any vector can be moved parallel to itself without being These four vectors are equal because they affected have equal lengths and point the same direction
Vectors and Their Properties
A vector is defined by its magnitude and direction. Therefore all of these vectors are the same, even though they are at different locations on the graph.
Vectors and Their Properties Adding Vector When adding vectors, their directions must be taken into account Units must be the same Geometric Methods Use scale drawings or graphically Algebraic Methods The resultant vector (sum) is denoted as
Vectors and Their Properties Adding Vectors Geometrically (Triangle or Polygon Method) Choose a scale Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system Draw the next vector using the same scale with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of the first vector and parallel to the ordinate system used for the first vector Continue drawing the vectors “tip-to-tail” The resultant is drawn from the origin of the first vector (tail) to the end of the last vector (tip)
Vectors and Their Properties
Adding Vectors Geometrically (Triangle or Polygon Method) Measure the length of the resultant and its angle Use the scale factor to convert length to actual magnitude This method is called the triangle method or tail to tip method
Vectors and Their Properties Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last.
Vectors and Their Properties
Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.
The resultant is still drawn from the origin of the first vector to the end of the last vector (many vectors)
Vectors and Their Properties Vector Addition Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn’t affect the result m
Vectors and Their Properties More Properties of Vectors Negative Vectors The negative of the vector A is defined as the vector that gives zero when added to the original vector A . Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) m
Vectors and Their Properties Vector Subtraction The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Special case of vector addition Add the negative of the subtracted vector Here,
Vectors and Their Properties Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar If the scalar is positive, the direction of the result is the same as of the original vector
3( A B) 3 A 3 B If the scalar is negative, the direction of the result is opposite that of the original vector
3( A B) 3 A 3 B
Vectors and Their Properties Multiplying a Vectors by a Scalars: the multiplier changes the length and the sign indicates the direction.
Components of a Vector
Resolve vector into perpendicular components using a two-dimensional coordinate system:
These are the projections of the vector along the x-axes and y-axes
Components of a Vector ď ą The x-component of a vector is the projection along the x-axis ď ą The y-component of a vector is the projection along the y-axis
Components of a Vector
θ is measured with respect to the x-axis
Given the magnitude and direction of a vector. Find its components: Ax A cos q Ay A sin q Given the components of a vector. Find its magnitude and direction: Ay 1 2 2 A Ax Ay q tan Ax
Components of a Vector The x-component of a vector is the projection along the x-axis Ax A sin q The y-component of a vector is the projection along the y-axis Ay A cos q
Components of a Vector
θ is measured with respect to the y-axis
Given the magnitude and direction of a vector. Find its components: Ax A sin q Ay A cos q Given the components of a vector. Find its magnitude and direction: Ax 2 2 q tan 1 A Ax Ay Ay
Components of a Vector Other Coordinate Systems It may be convenient to use a coordinate system other than horizontal and vertical Choose axes that are perpendicular to each other Adjust the components accordingly
The components of vector in a tilted B system. coordinate
Components of a Vector Length, angle and components can be calculated from each other using trigonometry.
Components of a Vector Signs of vector components:
Components of a Vector Signs of vector components: y
B Bx -ve By +ve Cx -ve Cy -ve
C
A Ax +ve Ay +ve Dx +ve Dy -ve
D
x
Components of a Vector
Adding Vectors Algebraically Choose a coordinate system and sketch the vectors Find the x-component and y-components of all the vectors Add all the x-components: This gives Rx Add all the y-components: This gives Ry Use the Pythagorean Theorem to find the magnitude of the resultant: Use the inverse tangent function to find the direction of R:
Components of a Vector
Unit vectors are dimensionless vectors of unit length
* Vector Multiplication There are two distinct ways to multiply vectors, referred to as the dot (scalar) product and the cross (vector) product. The difference between these two types of multiplication is that the dot product yields a scalar (a number) as its result, whereas the cross product results in a vector. Both types of product have important applications in physics.
* Scalar or Dot Product Definition of the scalar or dot product:
Therefore, we can write:
* Scalar or Dot Product
Example:
Scalar product equals: = Vectors are perpendicular between each other. Therefore,
* Vector or Cross Product The vector or cross product is defined as:
Some properties of the cross product:
One application of cross products is torque.
* Vector or Cross Product Example:
The cross product can also be written in determinant form: