James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

812 Chapter 12 Vectors and the Geometry of Space
But then, from Theorem 3, we have
12 W − | F | | D | cos − F ? D
Thus the work done by a constant force F is the dot product F ? D, where D is the displacement
vector.
F
35°
D
FIGURE 7
35°
ExamplE 7 A wagon is pulled a distance of 100 m along a horizontal path by a
constant force of 70 N. The handle of the wagon is held at an angle of 358 above the
horizontal. Find the work done by the force.
SOLUTION If F and D are the force and displacement vectors, as pictured in Figure 7,
then the work done is
W − F ? D − | F | | D | cos 35°
− s70ds100d cos 35° < 5734 N∙m − 5734 J
ExamplE 8 A force is given by a vector F − 3i 1 4j 1 5k and moves a particle from
the point Ps2, 1, 0d to the point Qs4, 6, 2d. Find the work done.
SOLUTION The displacement vector is D − PQ l − k 2, 5, 2 l, so by Equation 12, the
work done is
W − F ? D − k 3, 4, 5 l ? k 2, 5, 2 l
− 6 1 20 1 10 − 36
■
If the unit of length is meters and the magnitude of the force is measured in newtons,
then the work done is 36 J.
■
1. Which of the following expressions are meaningful? Which are
meaningless? Explain.
(a) sa ? bd ? c
(b) sa ? bdc
(c) | a | sb ? cd
(d) a ? sb 1 cd
(e) a ? b 1 c
? sb 1 cd
2–10 Find a ? b.
2. a − k5, 22l, b − k3, 4l
3. a − k1.5, 0.4l, b − k24, 6l
(f) | a |
11–12 If u is a unit vector, find u ? v and u ? w.
11.
12.
u
v
v
w
w
u
4. a − k 6, 22, 3 l, b − k 2, 5, 21 l
5. a − k4, 1, 1 4 l, b − k 6, 23, 28 l
6. a − k p, 2p, 2p l, b − k 2q, q, 2q l
7. a − 2i 1 j, b − i 2 j 1 k
8. a − 3i 1 2j 2 k, b − 4i 1 5k
9. | a | − 7, | b |
− 4, the angle between a and b is 30°
10. | a | − 80, | b | − 50, the angle between a and b is 3y4
13. (a) Show that i ? j − j ? k − k ? i − 0.
(b) Show that i ? i − j ? j − k ? k − 1.
14. A street vendor sells a hamburgers, b hot dogs, and c soft
drinks on a given day. He charges $4 for a hamburger, $2.50
for a hot dog, and $1 for a soft drink. If A − k a, b, c l and
P − k 4, 2.5, 1 l, what is the meaning of the dot product A ? P ?
15–20 Find the angle between the vectors. (First find an exact
expression and then approximate to the nearest degree.)
15. a − k 4, 3 l, b − k 2, 21 l
16. a − k 22, 5 l, b − k 5, 12 l
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