Thomas Calculus 13th [Solutions]

Chapter 12 Additional and Advanced Exercises 925
i j k i j k
v w 1 0 1 4i 4j 4 k u ( v w) 1 1 2 12i 4j 8 k;
2 4 2 4 4 4
( u w) v ( u v) w 10 i k 1 2i 4j 2k 12i 4j 8k
18. (a) u ( v w) v ( w u) w ( u v) ( u w) v ( u v) w ( v u) w ( v w) u ( w v) u ( w u)
v 0
(b) u ( v i) i ( u ( v j) j ( u ( v k) k ( u v) i i ( u v) j j ( u v)
k k u v
(c) ( u v) ( w r) u v ( w r) u ( v r) w ( v w) r ( u w)( v r) ( u r)( v w)
u w
u r
v w
v r
19. The formula is always true; u u ( u v) w u ( u v) u ( u u) v w ( u v) u u ( u u)
u v w
2 2
| u | u v w | u | u v w
20. If u (cos B) i (sin B)
j and v (cos A) i (sin A) j , where A B , then u v | u | | v | sin ( A B)
k
i j k
cos B sin B 0 (cos Bsin A sin B cos A) k sin( A B) cos B sin A sin B cos A,
since
cos A sin A 0
| u| 1 and | v| 1.
2 2 2 2
21. If u ai bj and v ci dj , then u v u v cos ac bd a b c d cos
2 2 2 2 2 2 2 2 2 2 2
2
ac bd a b c d cos ac bd a b c d , since cos 1.
22. If u ai bj ck , then
2 2 2
u u a b c 0 and u u 0 iff a b c 0.
23.
2 2 2
2
| u v | u v u v u u 2 u v v v | u| 2 | u || v | | v | | u | | v | | u v | | u | | v |
24. Let denote the angle between w and u, and the angle between w and v. Let a | u| and b | v|.
Then cos
a b a b a b a v u ba
w u v u u v u u u v u u u
v u ba , and likewise, cos u v ba .
| w|| u| | w|| u| | w|| u| | w|| u| | w| a | w|
| w|
Since the angle between u and v is always and cos
2
cos , we have that w bisects the angle
between u and v.
25. | u| v | v| u | v| u | u| v | u | v | v | u | v | u | v | u | u | v | u | v | v | u | u | v
2 2 2 2 2 2
| v | u | u | v | v | u u | u | v v | v | u | u | v | v | | u | | u | | v | 0
2
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