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

128 D CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
49. (a - b) x (a + b) = (a - b) x a + (a - b) x b
=ax a+ (-b)x a +ax b+(- b)x b
=(a X a)- (b x a)+ (a X b)- (b x b)
= 0 - (b x a) + (a x b)- 0
=(ax b)+ (a x b)
= 2(a x b)
by Property 3 of Theorem 11
.
by Property 4 ofTheorem I I
by Property 2 ofTheorem II (with c =: -1)
by Example2
by Property I of Theorem ll
51. a X (b X c) + b X (c X a) + c X (a X b)
= [(a · c)b- (a · b)c] + [(b · a)c- (b · c)a] + [(c · b)a - (c: a)b]
= (a · c)b - (a· b)c+ (a · b)c- (b · c)a+ (b · c)a- (a · c)b = 0
by Exercise 50
53. (a) No. If a · b = a · c, then ·a · (b - c) = 0, so a is perpendicular to b - c, which can happen if b -:f. c. For example,
let a = (1, 1, 1), b = (1, 0, 0) and c = (0, 1, 0).
(b) No. If a x b =ax c then a x (b- c) = 0, which implies that a is parallel to b- c; which of course can happen
ifb -:f. c.
(c) Yes. Since a· c = a· b , a is perpendicular to b - c, by part (a). From part (b), a is also parallel to b - c. Thus since
a -:f. 0 but is both parallel and perpendicular to b - c, we have b - c = 0, so b = c.
12.5 Equations of Lines and Planes
1. (a) True; each of the first two lines has a direction vector parallel to the direction vector of the third line, so these vectors are
each scalar multipl~s of the third direction vector. Then the first two direction vectors are also scalar multiples of each
other, so these vectors, and hence the two lines, are parallel.
(b) False; for example, the x- and y-axes are both perpendicular to the z-axis, yet the x- andy-axes are not parallel.
(c) True; each of the first two planes has a normal vector parallel to ~he normal vector of the third plane, so these two normal
vectors are parallel to each other and the planes are parallel.
(d) False; for example, the xy- and yz-planes are not parallel, yet they are both perpendicular to the xz-plane.
(e) False; the x- and y-axes are not parallel, yet they are both parallel to the plane z = 1.
(f) True; if each line is perpendicular to a plane, then the lines' direction vectors are both parallel to a normal vector for the
plane. Thus, the direction vectors arc parallel to each other and the lines are parallel.
(g) False; the planes y = 1 and z = 1 are not parallel, yet they are both parallel to the x-axis.
(h) True; if each plane is perpendicular to a line; then any normal vector for each plane is parallel to a direction vector for the
line. Thus, the normal vectors are parallel to each other and the planes are parallel.
(i) True; see Figure 9 and the accompanying discussion.
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