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

124 0 CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE
j k
5. ax b = 1 - 1 -1
= ,-~ -~ li 11 - 1, . 11
1 1 J + 1 -~ lk
1
1 2 2 2
2 1 2
Now {ax b)· a= {~ i-j + ~ k ) · (i-j - k) = ~ + 1- ~
~ 0 and
(a x b ) · b = { ~ i - j + ~ k) · G i + j + ~ k) = ~ - 1 + ~ = 0, so a x b is orthogonal to both a and b .
j k
= 1 1/t I
i -I t 1/t I· + I t
7. a x b = t 1 1/t
1 I k
t21 t21 J t
e 2 t2
t 2 1
= (1 - t) i - (t- t) j + (t 3 - t 2 ) k = (1 - t) i + (t 3 - t 2 ) k
Since {a X b). a = (1 -t,O,t 3 - t2). (t, 1, 1/t) = t-e +0 +t 2 - t = 0, a X b is orthogonal to a.
Since (a x b ) · b = ( 1 - t, 0, t 3 - t 2 ) • ( t 2 , t 2 , 1) = t 2 - t 3 + 0 + t 3 - t 2 = 0, a x b is orthogonal to b .
9. According to the discussion precedin.g Theorem II, i x "j = k, so (i x j ) x k = k x k = 0 [by Example 2).
11. (j - k) X (k- i) = (j - k) X k + (j - k) .X ( - i)
= j X k + (- k) X k + j X (-i) + (-k) X (- i)
= (j X k ) + (-1){k X k) + (- 1)(j Xi)+ (-1) 2 (k X i)
= i + (-1) 0 + (-1)(- k) + J = i + j + k
by Property 3 of Theorem I I
by Property 4 of Theorem 11
by Property 2 of Theorem I 1
by Example 2 and
the discussion preceeding Theorem II
13. (a) Since b x cis a vector, the dot product a· (b x c) is meaningful and is a scalar.
(b) b ·cis a scalar, so a x (b ·c) is meaningless, as the cross product is defined only for two vectors.
(c) Since b x cis a vector, the cross'product a x (b x c) is meaningful and results in another vector.
(d) b · cis a scalar, so the dot product a · (b ·c) is meaningless, as the dot product is defined only for two vectors.
(e) Since (a . b ) and ( c . d) are both scalars, the cross product (a . b ) X ( c . d) is meaningless.
(f) a x band c x d arc both vectors, so the dot product (ax b)· (c x d ) is meaningful and is a scalar.
15. If we sketch u an~ v starting from the same initial point, we see.that the
angle between them is 60°. Using Theorem 9, we have
lu x vi = lullvl sin 0 = {12)(16) sin 60° = 192 · v; = 96 v'3.
By the right-hand rule, u x v is directed into the page. .
v
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