Thomas Calculus 13th [Solutions]

914 Chapter 12 Vectors and the Geometry of Space
20. projvu v u v 1
| v|| v| 3
i 2j where v u 1 and v v 3
i j k
21. u v 1 0 0 k
1 1 0
i j k
22. u v 1 1 0 2k
1 1 0
23. Let v v1i v2j v3k and w w 1 i w 2 j w 3 k . Then 2
2
v 2w v1i v2j v3k 2 w1i w2 j w3k
2
2 2 2
2
v1 2w1 i v2 2w2 j v3 2w3 k v1 2w1 v2 2w2 v3 2w3
2 2 2 2 2 2 2 2
v1 v2 v3 4 v1w1 v2w2 v3w3 4 w1 w2 w3
| v| 4v w 4 | w|
2 2
| v | 4 | v | | w | cos 4 | w | 4 4(2)(3) cos 36 40 24 1 40 12 28 | v 2 w | 28 2 7
3 2
i j k
24. u and v are parallel when u v 0 2 4 5 0 (4a
40) i (20 2 a) j (0) k 0
4 8 a
4a 40 0 and 20 2a a 10
i j k
25. (a) area | u v | abs 1 1 1 2i 3j k 4 9 1 14
2 1 1
1 1 1
(b) volume ( u v) w 2 1 1 1(3 2) 1 6 ( 1) 1( 4 1) 1
1 2 3
i j k
26. (a) area | u v | abs 1 1 0 | k | 1
0 1 0
1 1 0
(b) volume ( u v) w 0 1 0 1(1 0) 1(0 0) 0 1
1 1 1
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