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Unformatted text preview: k z 2 sin ± 30 k 0, 3, ± 1 3, 2, 2, 6, 3 4, 2 1, 2 4, 3, 0 −2 2, 4, 2 −2 1 −1 1 −1 −2 0, 3, 1 2 x y 0, 3, −1 99. (a) 1 z (b) w a v au 0, a bv b ai 0, b a 0 bj bk 0 Thus, a and b are both zero. 1 y 1 x u (c) ai a w 101. d a 1, b u x2 bj 1 v x1 2 bk i 2j k (d) a i a a 1, a bj b bk 2, b i 3 2j 3k Not possible y2 y1 2 z2 z1 2 103. Two nonzero vectors u and v are parallel if u for some scalar c. L2 182. cv 105. (a) The height of the right triangle is h The vector PQ is given by \ \ Q (0, 0, h) PQ 0, 18, h . L The tension vector T in each wire is 24 T c 0, 18, h where ch 3 8 Hence, T 0, 18, h and h T (b) L T T 20 18.4 8 h 182 25 11.5 h2 30 10 8 L2 35 9.3 8. (0, 0, 0) 18 (0, 18, 0) P 182 40 9.0 182 45 8.7 L2 50 8.6 182 8L L2 182 —CONTINUED— S ection 10.2 105. —CONTINUED— (c) 30 Space Coordinates and Vectors in Space 237 107. Let v be the angle between v and the coordinate axes. cos i cos 1 3 3 j k 3 1, 1, 1 3 j cos k L = 18 v cos T=8 0 0 100 3 cos 1 3 v x 18 is a vertical asymptote and y horizontal asymptote. (d) lim 8L L2 8L L2 182 182 L→ L → 18 8 is a z 0.6 0.4 0.2 3 i 3 ( 3 , 3 3 , 3 3 3 ( y L→ lim lim 8 1 18 L 2 8 0.6 x 0.2 0.4 0.4 (e) From the table, T \ 10 implies L 30 inches. 109. AB \ 0, 70, 115 , F1 60, 0, 115 , F2 45, F1 F2 C1 0, 70, 115 C2 60, 0, 115 65, 115 AC \ AD F 65, 115 , F3 F3 C3 45, 0, 0, 500 Thus: 60C2 70C1 115 C1 C2 45C3 65C3 C3 500 104 69 , 0 0 Solving this system yields C1 F1 F2 F3 111. d AP x2 x2 y2 202.919N 157.909N 226.521N C2 28 23 , and C3 112 69 . Thus: 2d BP y z2 1 2 z 2z 1 2 2 4 x2 3x2 x2 x x 1 y2 3y2 8 x 3 4 3 2 2 y z2 3z2 16 9 y 3 2x 2 2 z2 4y 18y 6y z 1 3 5 2z 9 2 2y 2 0 8x y2 2 18 z2 2 z 3 1 9 6 16 9 9 1 9 44 9 Sphere; center: 4 , 3, 3 1 2 11 , radius: 3 3 238 Chapter 10 Vectors and the Geometry of Space Section 10.3 1. u (a) u (b) u (c) u (d) u (e) u 5. u 2i v u 2 2 The Dot Product of Two Vectors 2, 4 44 3 3 25 6 3. u (a) u (b) u (c) u 6 2, 2u v i 3 2 k 10 1 1 1 1 11 1 6 u 6 12, 18 12 7. (d) u (e) u u v v 2 3, 4 , v v u 32 33 25 vv 2v j 2, v u 3, 4 , v 20 22 29 vv 2v 0, 6, 5 36 3 3 45 44 2 29 2 0, 6, 5 2u v 0, 12, 10 22 4 k, v 21 22 6 3240, 1450, 2235 2.22, 1.85, 3.25 $17,139.05 (a) u (b) u (c) u (d) u (e) u uv uv u v vv 2v v 2u i v k 2 This gives the tota...
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