U1 v u 2w 4 2 u1 u2 3 3 5 1 u2 v 2w 2 x u 2 4 6 q 3

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Unformatted text preview: 3, 5 31. v 16 9 5 33. v 36 25 61 35. v 0 16 4 37. u v u u 32 122 3, 12 153 153 3 , 153 12 153 unit vector 39. u v u u 3 2 2 5 2 2 34 2 3 , 34 5 34 3 2, 5 2 34 2 17 4 17 , 17 17 3 34 5 34 , unit vector 34 34 Section 10.1 1 ,v 2 1 4 v v u u u u (e) 0, 1 1 (f) u u v v u u v v v v 1 v v 1 1 1 2, 3 13 9 3, 7 2 9 2 1 1, 2 5 49 4 Vectors in the Plane 229 41. u (a) u (b) v (c) u u (d) 1, 1 ,v 1 1 v v 1 4 0, 1 0 1 1, 2 1 1, 2 2 5 43. u (a) u (b) v 1, 2, 3 1 4 5 2 13 1 1 1 (c) u u u u u u 85 2 (e) v v (f) u u v v 1 5 1 (d) 1, 2 u u v v v v 7 2 3, 2 85 45. u u v v u u u v v 2, 1 5 5, 4 41 7, 5 74 8.602 v 6.403 2.236 47. 4 u u u u v 1 1, 1 2 2 2 1, 1 2 2, 2 2 v≤u 49. 2 u u u u v 1 23 1 3 1, 3 3, 3 3, 3 51. v 3 cos 0 i sin 0 j 3i 3, 0 53. v 2 cos 150 i 3i j sin 150 j 3, 1 55. u v u v i 32 i 2 2 32 j 2 32 j 2 32 i 2 230 57. Chapter 10 u v u v 2 cos 4 i cos 2 i 2 cos 4 Vectors and the Geometry of Space 2 sin 4 j sin 2 j cos 2 i 2 sin 4 sin 2 j 59. A scalar is a real number. A vector is represented by a directed line segment. A vector has both length and direction. 61. To normalize v, you find a unit vector u in the direction of v: u v . v bw ai 2j bi j a bi 2a b j. For Exercises 63–67, au 63. v 2i j. Therefore, a b 2, 2a simultaneously, we have a 1, b 1. 67. v i j. Therefore, a b simultaneously, we have a 69. y x 3, y 3x 2 3. Let w 3 at x 1, 2a b b 1. 3 b 1. Solving 65. v 3i. Therefore, a b 3, 2a simultaneously, we have a 1, b b 2. 0. Solving 1. Solving 2 3, 1. 71. f x fx (a) m 25 x 25 3 4. x2 x2 Let w 4, 3 . 3, 4 , then 3 at x 4 3. (a) m w w (b) m w w 1, 3 , then 1 1, 3 . ± 10 1 3. 4, 3 , then Let w 3, 1. 1 , then 1 ± 3, 10 w w (b) m w w 4 3. ± 1 5 Let w ± 1 3, 4 5 73. u u v v 2 i 2 2j u v 2 j 2 75. Programs will vary. u 2 i 2 2 j 2 77. F1 F2 F3 R R 2, 3, F1 F2 F3 33 125 110 F3 132.5 1.33 2.5, F1 F1 F2 F2 F3 79. (a) 180 cos 30i Direction: Magnitude: sin 30j arctan 430.882 275i 90 430.88 902 430.88 i 0.206 90 j 11.8 (b) M 275 arctan 180 cos 275 2 180 sin 2 180 sin 180 cos 4...
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