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
21 31
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
a
14 393
,
,
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.
l
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
l
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
a
s14 a F cos ) D But then, from Theorem 3, we have
12 W F D cos FD 5E13(pp 848857) ❙❙❙❙ 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.
F
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.
l
SOLUTION The displacement vector is D
PQ
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
 hamrick
 Linear Algebra, Vectors, Vector Space, Dot Product, (pp 828837)

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