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Unformatted text preview: LE 6 Find the scalar projection and vector projection of b a 1, 1, 2 onto 2, 3, 1 . SOLUTION Since a s
compa b 2 2 32 12 ab
a s14, the scalar projection of b onto a is
s14 12 3
s14 The vector projection is this scalar projection times the unit vector in the direction of a:
proja b F P Q
D ( W S ¨ 3
7 14 14 One use of projections occurs in physics in calculating work. In Section 6.4 we deﬁned
the work done by a constant force F in moving an object through a distance d as W Fd,
but this applies only when the force is directed along the line of motion of the object.
Suppose, however, that the constant force is a vector F PR pointing in some other direction as in Figure 6. If the force moves the object from P to Q, then the displacement
vector is D PQ. The work done by this force is deﬁned to be the product of the component of the force along D and the distance moved: R FIGURE 6 3
s14 a F cos ) D But then, from Theorem 3, we have
12 W F D cos FD 5E-13(pp 848-857) ❙❙❙❙ 848 1/18/06 11:18 AM Page 848 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE Thus, the work done by a constant force F is the dot product F D, where D is the displacement vector.
EXAMPLE 7 A crate is hauled 8 m up a ramp under a constant force of 200 N applied at
an angle of 25 to the ramp. Find the work done.
25° SOLUTION If F and D are the force and displacement vectors, as pictured in Figure 7, then
the work done is D W
FIGURE 7 FD F D cos 25 200 8 cos 25 1450 N m 1450 J EXAMPLE 8 A force is given by a vector F
3 i 4 j 5 k and moves a particle from
the point P 2, 1, 0 to the point Q 4, 6, 2 . Find the work done.
SOLUTION The displacement vector is D
2, 5, 2 , so by Equation 12, the work
done is W FD 3, 4, 5 6 10 20 2, 5, 2
36 If the unit of length is meters and the magnitude of the force is measured in newtons,
then the work done is 36 joules. |||| 13.3 Exercises 1. Which of the following expressions are meaningful? Which are meaningless? Explain.
(a) a b c
(c) a b c
(e) a b c 11–12 |||| If u is a...
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- Fall '09