Math 151
Section 1.2
The Dot Product
Work The work W done by a constant force F in moving an object through a distance d is given by
W = Fd. This formula only applies when the force is directed along the line of motion of the object.
Suppose we have an object moving from point P to point Q under a force F as shown in the figure.
F
P
θ
Q
S
The work done in moving the object from P to Q depends on two things:
!!!"
1. The distance the object has been moved, given by D = PQ , where D is the displacement vector
!!"
2. The magnitude of the force applied in the direction of D, given by PS = F cos!
So, the work done by moving the object is given by W = D F cos!
Example: Find the work done by a force of 10 N acting in the direction N50°W in moving an object
4 m due west.
The Dot Product The dot product of vectors a and b is given by
a !b = a b cos!
where θ is the angle between a and b, and 0 ! ! ! " . If a = 0 or b = 0, then a !b = 0 .
Example: Prove a ! b " a #b = 0 .
Math 151
The Dot Product Version 2 The dot product of vectors a = a1 , a2 and b = b1 ,b2 is given by
a !b = a1b1 + a2 b2
Properties of the Dot Product are given on p. 57 of the textbook.
Example: Calculate the dot product of 4i and –3j.
Example: A force given by the vector F = 3,8 moves an object along a straight line from point
(2, 3) to point (4, 5). Calculate the work done if the distance is measured in meters and the
magnitude of the force is measured in Newtons.
Example: Determine whether the given vector pairs are parallel, perpendicular, or neither.
A. 2,6 , !3,1
B. 2,6 , 3,9
Math 151
Orthogonal Complement The orthogonal complement of a vector a = a1 , a2 is given by
a ! = "a2 , a1 .
Example: Find a unit vector perpendicular to 2,!3 .
Vector and Scalar Projections Given vectors a and b,
The scalar projection of b onto a is: compa b =
The vector projection of b onto a is: proja b =
a !b
a
a !b
a
2
a
Example: Find the scalar and vector projections of 4,8 onto 2,1 .
Math 151
Example: Find the distance from the point P(2, 1) to the line y = 2x + 1.