# ODD10 - CHAPTER 10 Vectors and the Geometry of Space...

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C H A P T E R 1 0 Vectors and the Geometry of Space Section 10.1 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . 227 Section 10.2 Space Coordinates and Vectors in Space . . . . . . . . . . 232 Section 10.3 The Dot Product of Two Vectors . . . . . . . . . . . . . . 238 Section 10.4 The Cross Product of Two Vectors in Space . . . . . . . . 241 Section 10.5 Lines and Planes in Space . . . . . . . . . . . . . . . . . 244 Section 10.6 Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . 249 Section 10.7 Cylindrical and Spherical Coordinates . . . . . . . . . . . 252 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

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227 C H A P T E R 1 0 Vectors and the Geometry of Space Section 10.1 Vectors in the Plane Solutions to Odd-Numbered Exercises 1. (a) (b) 5 4 3 2 1 1 3 2 4 5 x v (4, 2) y v 5 1, 3 1 4, 2 3. (a) (b) 4 2 2 2 4 4 6 8 x v ( 7, 0) y v 4 3, 2 2 7, 0 5. u v v 1 1 , 8 4 2, 4 u 5 3, 6 2 2, 4 7. u v v 9 3, 5 10 6, 5 u 6 0, 2 3 6, 5 9. (b) (a) and (c). 4 4 2 2 x v (5, 5) (4, 3) (1, 2) y v 5 1, 5 2 4, 3 11. (b) (a) and (c). 10 2 4 6 2 4 x v y ( 4, 3) (6, 1) (10, 2) v 6 10, 1 2 4, 3 13. (b) (a) and (c). 6 4 6 4 2 2 x v (6, 6) (0, 4) (6, 2) y v 6 6, 6 2 0, 4 15. (b) (a) and (c). 2 1 3 2 1 2 x 5 , 1 v 3 ( ( 1 , 3 2 ( ( 3 , 2 4 3 ( ( y v 1 2 3 2 , 3 4 3 1, 5 3 17. (a) —CONTINUED— 6 6 4 2 2 4 x v v 2 (4, 6) (2, 3) y 2 v 4, 6 (b) 4 4 4 4 8 8 x v 3 v (2, 3) ( 6, 9) y 3 v 6, 9
17. —CONTINUED— (c) 12 8 8 4 4 12 2 x 7, (2, 3) ( ( v v 21 2 7 y 7 2 v 7, 21 2 (d) 3 2 1 1 2 3 x v v (2, 3) y 4 , 2 3 2 3 ( ( 2 3 v 4 3 , 2 19. x u y 21. x v u u v y 23. (a) (b) (c) 2 u 5 v 2 4, 9 5 2, 5 18, 7 v u 2, 5 4, 9 2, 14 2 3 u 2 3 4, 9 8 3 , 6 25. 3 2 1 1 2 3 3 2 3 2 v = u x u u y 3, 3 2 v 3 2 2 i j 3 i 3 2 j 27. 4 i 3 j 4, 3 4 2 4 2 6 v v = u + 2 w w 2 x u y v 2 i j 2 i 2 j 29. Q 3, 5 u 2 5 u 1 3 u 2 2 3 u 1 4 1 31. v 16 9 5 33. v 36 25 61 35. v 0 16 4 37. unit vector 17 17 , 4 17 17 v u u 3, 12 153 3 153 , 12 153 u 3 2 12 2 153 39. unit vector 3 34 34 , 5 34 34 v u u 3 2 , 5 2 34 2 3 34 , 5 34 u 3 2 2 5 2 2 34 2 228 Chapter 10 Vectors and the Geometry of Space

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41. (a) (b) (c) (d) (e) (f) u v u v 1 u v u v 0, 1 v v 1 v v 1 5 1, 2 u u 1 u u 1 2 1, 1 u v 0 1 1 u v 0, 1 v 1 4 5 u 1 1 2 u 1, 1 , v 1, 2 43. (a) (b) (c) (d) (e) (f) u v u v 1 u v u v 2 85 3, 7 2 v v 1 v v 1 13 2, 3 u u 1 u u 2 5 1, 1 2 u v 9 49 4 85 2 u v 3, 7 2 v 4 9 13 u 1 1 4 5 2 u 1, 1 2 , v 2, 3 45. u v u v u v 74 8.602 u v 7, 5 v 41 6.403 v 5, 4 u 5 2.236 u 2, 1 47. v 2 2 , 2 2 4 u u 2 2 1, 1 u u 1 2 1, 1 49. v 1, 3 2 u u 1 3 3 , 3 u u 1 2 3 3 , 3 51. v 3 cos 0 i sin 0 j 3 i 3, 0 53. 3 i j 3 , 1 v 2 cos 150 i sin 150 j 55. u v 2 3 2 2 i 3 2 2 j v 3 2 2 i 3 2 2 j u i Section 10.1 Vectors in the Plane 229
57. u v 2 cos 4 cos 2 i 2 sin 4 sin 2 j v cos 2 i sin 2 j u 2 cos 4 i 2 sin 4 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 , you find a unit vector in the direction of u v v . v : u v For Exercises 63–67, a u b w a i 2j b i j a b i 2 a b j. 63. Therefore, Solving simultaneously, we have b 1. a 1, 2 a b 1. a b 2, v 2 i j . 65. Therefore, Solving simultaneously, we have a 1, b 2. 2 a b 0. a b 3, v 3 i . 67. Therefore, Solving simultaneously, we have a 2 3 , b 1 3 .

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