Thomas Calculus 13th [Solutions]

924 Chapter 12 Vectors and the Geometry of Space
14. Let n AB BC and D( x, y, z ) be any point in the plane determined by A, B and C. Then the point D lies in
this plane if and only if AD n 0 AD AB BC 0.
i j k
15. n i 2j 6k is normal to the plane x 2y 6z 6; v n 1 1 1 4i 5j k is parallel to the plane and
1 2 6
i j k
perpendicular to the plane of v and n w n ( v n) 1 2 6 32i 23j 13k is a vector parallel to the
4 5 1
plane x 2y 6z 6 in the direction of the projection vector proj P v . proj
32 23 13 42 1 32 23 13
Pv proj w w v w
w v v w w w w i j k
| w| | w| 2 2 2 2
| w| 32 23 13 1722 41 41 41 41
16. projz w projz v and w projz w v projz v w w projz w projz w v projz v projz
w
v 2 projz
v v 2 v z z
2
| z|
17. (a) u v 2i 2j 4 k ( u v) w 0;( u w) v ( v w) u 0v 0 u 0; v w 4 i u ( v w) 0;
( u w) v ( u v) w 0v 0w 0
i j k i j k
(b) u v 1 1 1 i 4j 3 k ( u v) w 1 4 3 10i 2j 6 k;
2 1 2 1 2 1
( u w) v ( v w) u 4 2i j 2k 2 i j k 10i 2j 6 k;
i j k i j k
v w 2 1 2 3i 4j 5 k u ( v w) 1 1 1 9i 2j 7 k;
1 2 1 3 4 5
( u w) v ( u v) w 4 2i j 2 k ( 1) i 2j k 9i 2j 7k
i j k i j k
(c) u v 2 1 0 i 2j 4 k ( u v) w 1 2 4 4i 6j 2 k;
2 1 1 1 0 2
( u w) v ( v w) u 2 2i j k 4 2i j 4i 6j 2 k;
i j k i j k
v w 2 1 1 2i 3 j k u ( v w) 2 1 0 i 2j 4 k;
1 0 2 2 3 1
( u w) v ( u v) w 2 2i j k 3 i 2k i 2j 4k
i j k i j k
(d) u v 1 1 2 i 3 j k ( u v) w 1 3 1 10i 10 k;
1 0 1 2 4 2
( u w) v ( v w) u 10 i k 0 i j 2k 10i 10 k;
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