Thomas Calculus 13th [Solutions]

23. (a) u v 6, u w 81, v w 18 none are perpendicular
i j k i j k i j k
Section 12.4 The Cross Product 895
(b) u v 5 1 1 0, u w 5 1 1 0, v w 0 1 5 0 u and w are parallel
0 1 5 15 3 3 15 3 3
24. (a) u v 0, u w 0, u r 3 , v w 0, v r 0, w r 0 u v, u w, v w,
v r and w r
(b)
i j k i j k i j k
u v 1 2 1 0, u w 1 2 1 0, u r 1 2 1 0
1 1 1 1 0 1
2 2
i j k i j k i j k
v w 1 1 1 0, v r 1 1 1 0, w r 1 0 1 0 u and r are parallel
1 0 1
2 2 2 2
25.
PQ
F
2 3
PQ | F| sin(60 ) 30 ft lb 10 3 ft lb
3 2
26.
PQ
F
2 2
PQ | F| sin(135 ) 30 ft lb 10 2 ft lb
3 2
27. (a) true, | u| 2
a1 2
a2 2
a3
u u
(b) not always true, u u
2
| u|
i j k
i j k
(c) true, u 0 u1 u2 u3 0i 0j 0k 0 and 0 u 0 0 0 0i 0j 0k 0
0 0 0
u1 u2 u3
i j k
(d) true, u ( u) u1 u2 u3 u2u3 u2u3 i u1u 3 u1u 3 j u1u 2 u1u
2 k 0
u1 u2 u3
(e) not always true, i j k k j i for example
(f ) true, distributive property of the cross product
(g) true, ( u v) v u ( v v) u 0 0
(h) true, the volume of a parallelepiped with u, v , and w along the three edges is the same whether the plane
containing u and v or the plane containing v and w is used as the base plane, and the dot product is
commutative.
28. (a) true, u v u1v1 u2v2 u3v3 v1u 1 v2u2 v3u3
v u
i j k i j k
(b) true, u v u1 u2 u3 v1 v2 v3
( v u)
v1 v2 v3 u1 u2 u3
i j k i j k
(c) true, ( u) v u1 u2 u3 u1 u2 u3
( u v)
v1 v2 v3 v1 v2 v3
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