Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

142 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
17. (a) See Exercise 12.4.45.
(b) See Example 8 in Section 12.5.
(c) See Example 10 in Section 12.5.
18. The traces of a surface are the curves of intersection of the surface with planes parallel to the coordinate planes. We can find
the trace in the plane x = k (parallel to the yz-plane) by setting x = k and determining the curve represented by the resulting
equation. Traces in the planes y = k (parallel to the xz-plane) and z = k {parallel to the xy-plane) are found similarly.
19. See Table I in Section 12.6.
TRUE-FALSE QUIZ
1. This is false, as the dot product of two vectors is a scalar, not a vector.
3. False. For example, ifu = i and ·v = j then iu · v i= 101 = 0 but iu l lvl = 1 · 1 = 1. In fact, by Theorem 12.3.3,
lu · v i = liu llvl cos 91.
5. True, by Theorem 12.3.2, property 2.
7. True. If 9 is the angl~ between u and v , then by Theorem 12.4.9,lu x vi = iu l lvl sin 9 = lvllul sin 9 = 1':' x ui.
(Or, by Theorem 12.4.1l,lu x v i = 1-v x ul = 1- lllv x ul = lv x ui.)
9. Theorem 12.4.11 , property 2 tells us that this is true.
11. This is true by Theorem 12.4.11, property 5.
13. This is true because u x vis orthogonal to u (see Theorem 12.4.8), and the dot product of two orthogonal vectors is 0.
15. This is false. A normal vector to the plane is n = (6, - 2, 4). Because (3, - 1, 2) = ~n , the vector is parallel ton and hence
perpendicular to the plane.
17. This is false. In IR 2 , x 2 + y 2 = 1 represents a circle, but { (x, y, z) I x 2 + y 2 = 1} represents a three-dimensional surface,
namely, a circular cylinder with axis the z -axis.
19. False. For example, i · j = 0 but i i- 0 and j =f. 0.
21. This is true. Ifu and v are both nonzero, then by (7) in s~ c tion 12.3, u · v = 0 implies that u and v are orthogonal. But
u x v = 0 implies that u and v are parallel (see Corollary 12.4.10). Two nonzero vectors can't be both parallel and
orthogonal, so at least one of u, v must be 0.
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