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

Section 12.4 The Cross Product 821
SOLUTION The magnitude of the torque vector is
| t | − | r 3 F | − | r | | F | sin 75° − s0.25ds40d sin 75°
− 10 sin 75° < 9.66 N∙m
If the bolt is right-threaded, then the torque vector itself is
t − | t | n < 9.66 n
where n is a unit vector directed down into the page (by the right-hand rule).
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1–7 Find the cross product a 3 b and verify that it is orthogonal to
both a and b.
1. a − k2, 3, 0l, b − k1, 0, 5l
2. a − k4, 3, 22l, b − k2, 21, 1l
3. a − 2j 2 4k, b − 2i 1 3j 1 k
4. a − 3i 1 3j 2 3k, b − 3i 2 3j 1 3k
5. a − 1 2 i 1 1 3 j 1 1 4 k, b − i 1 2 j 2 3 k
6. a − ti 1 cos tj 1 sin tk, b − i 2 sin tj 1 cos tk
7. a − k t, 1, 1yt l, b − kt 2 , t 2 , 1 l
8. If a − i 2 2k and b − j 1 k, find a 3 b. Sketch a, b, and
a 3 b as vectors starting at the origin.
9–12 Find the vector, not with determinants, but by using
properties of cross products.
9. si 3 jd 3 k 10. k 3 si 2 2jd
11. s j 2 kd 3 sk 2 id 12. si 1 jd 3 si 2 jd
13. State whether each expression is meaningful. If not, explain
why. If so, state whether it is a vector or a scalar.
(a) a ? sb 3 cd
(b) a 3 sb ? cd
(c) a 3 sb 3 cd
(d) a ? sb ? cd
(e) sa ? bd 3 sc ? dd (f) sa 3 bd ? sc 3 dd
14–15 Find | u 3 v | and determine whether u 3 v is directed into
the page or out of the page.
14.
|u|=4
45°
|v|=5
15.
|u|=12
120°
|v|=16
16. The figure shows a vector a in the xy-plane and a vector b in
the direction of k. Their lengths are | a | − 3 and | b | − 2.
(a) Find | a 3 b | .
(b) Use the right-hand rule to decide whether the com ponents
of a 3 b are positive, negative, or 0.
x
z
b
17. If a − k2, 21, 3 l and b − k4, 2, 1 l, find a 3 b and b 3 a.
18. If a − k1, 0, 1 l, b − k2, 1, 21 l , and c − k0, 1, 3 l, show that
a 3 sb 3 cd ± sa 3 bd 3 c.
19. Find two unit vectors orthogonal to both k3, 2, 1 l and
k21, 1, 0 l.
20. Find two unit vectors orthogonal to both j 2 k and i 1 j.
21. Show that 0 3 a − 0 − a 3 0 for any vector a in V 3.
22. Show that sa 3 bd ? b − 0 for all vectors a and b in V 3.
23–26 Prove the property of cross products (Theorem 11).
23. Property 1: a 3 b − 2b 3 a
24. Property 2: scad 3 b − csa 3 bd − a 3 scbd
25. Property 3: a 3 sb 1 cd − a 3 b 1 a 3 c
26. Property 4: sa 1 bd 3 c − a 3 c 1 b 3 c
27. Find the area of the parallelogram with vertices As23, 0d,
Bs21, 3d, Cs5, 2d, and Ds3, 21d.
28. Find the area of the parallelogram with vertices Ps1, 0, 2d,
Qs3, 3, 3d, Rs7, 5, 8d, and Ss5, 2, 7d.
29–32 (a) Find a nonzero vector orthogonal to the plane through
the points P, Q, and R, and (b) find the area of triangle PQR.
29. Ps1, 0, 1d, Qs22, 1, 3d, Rs4, 2, 5d
30. Ps0, 0, 23d, Qs4, 2, 0d, Rs3, 3, 1d
31. Ps0, 22, 0d, Qs4, 1, 22d, Rs5, 3, 1d
a
y
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