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

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CHAPTER 10 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 CHAPTER 10 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 k 5 2 1, 3 2 1 l 5 k 4, 2 l 3. (a) (b) 4 2 2 2 4 4 6 8 x v ( 7, 0) y v 5 k 2 4 2 3, 2 2 2 s 2 2 d l 5 k 2 7, 0 l 5. u 5 v v 5 k 1 2 s 2 1 d , 8 2 4 l 5 k 2, 4 l u 5 k 5 2 3, 6 2 2 l 5 k 2, 4 l 7. u 5 v v 5 k 9 2 3, 5 2 10 l 5 k 6, 2 5 l u 5 k 6 2 0, 2 2 2 3 l 5 k 6, 2 5 l 9. (b) (a) and (c). 4 4 2 2 x v (5, 5) (4, 3) (1, 2) y v 5 k 5 2 1, 5 2 2 l 5 k 4, 3 l 11. (b) (a) and (c). 10 2 4 6 2 4 x v y ( 4, 3) (6, 1) (10, 2) v 5 k 6 2 10, 2 1 2 2 l 5 k 2 4, 2 3 l 13. (b) (a) and (c). 6 4 6 4 2 2 x v (6, 6) (0, 4) (6, 2) y v 5 k 6 2 6, 6 2 2 l 5 k 0, 4 l 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 5 k 1 2 2 3 2 , 3 2 4 3 l 5 k 2 1, 5 3 l 17. (a) —CONTINUED— 6 6 4 2 24 x v v 2 (4, 6) (2, 3) y 2 v 5 k 4, 6 l (b) 4 4 4 4 8 8 x v 3 v (2, 3) ( 6, 9) y 2 3 v 5 k 2 6, 2 9 l
17. —CONTINUED— (c) 12 8 8 4 4 12 2 x 7, (2, 3) ( ( v v 21 2 7 y 7 2 v 5 k 7, 21 2 l (d) 3 2 1 123 x v v (2, 3) y 4 , 2 3 2 3 ( 2 3 v 5 k 4 3 , 2 l 19. x u y 21. x v u uv y 23. (a) (b) (c) 2 u 1 5 v 5 2 k 4, 9 l 1 5 k 2, 2 5 l 5 k 18, 2 7 l v 2 u 5 k 2, 2 5 l 2 k 4, 9 l 5 k 2 2, 2 14 l 2 3 u 5 2 3 k 4, 9 l 5 k 8 3 , 6 l 25. 3 2 1 1 2 3 3 2 3 2 v = u x u u y 5 k 3, 2 3 2 l v 5 3 2 s 2 i 2 j d 5 3 i 2 3 2 j 27. 5 4 i 1 3 j 5 k 4, 3 l 4 2 4 2 6 v v = u + 2 w w 2 x u y v 5 s 2 i 2 j d 1 2 s i 1 2 j d 29. Q 5 s 3, 5 d u 2 5 5 u 1 5 3 u 2 2 2 5 3 u 1 2 4 52 1 31. i v i 5 ! 16 1 9 5 5 33. i v i 5 ! 36 1 25 5 ! 61 35. i v i 5 ! 0 1 16 5 4 37. unit vector 5 7 ! 17 17 , 4 ! 17 17 8 v 5 u i u i 5 k 3, 12 l ! 153 5 7 3 ! 153 , 12 ! 153 8 i u i 5 ! 3 2 1 12 2 5 ! 153 39. unit vector 5 7 3 ! 34 34 , 5 ! 34 34 8 v 5 u i u i 5 k s 3 ± 2 d , s 5 ± 2 d l ! 34 ± 2 5 7 3 ! 34 , 5 ! 34 8 i u i 5 ! 1 3 2 2 2 1 1 5 2 2 2 5 ! 34 2 228 Chapter 10 Vectors and the Geometry of Space

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41. (a) (b) (c) (d) (e) (f) i u 1 v i u 1 v i i 5 1 u 1 v i u 1 v i 5 k 0, 1 l i v i v i i 5 1 v i v i 5 1 ! 5 k 2 1, 2 l i u i u i i 5 1 u i u i 5 1 ! 2 k 1, 2 1 l i u 1 v i 5 ! 0 1 1 5 1 u 1 v 5 k 0, 1 l i v i 5 ! 1 1 4 5 ! 5 i u i 5 ! 1 1 1 5 ! 2 i u i 5 k 1, 2 1 l , v 5 k 2 1, 2 l 43. (a) (b) (c) (d) (e) i u 1 v i u 1 v i i 5 1 u 1 v i u 1 v i 5 2 ! 85 7 3, 7 2 8 i v i v i i 5 1 v i v i 5 1 ! 13 k 2, 3 l i u i u i i 5 1 u i u i 5 2 ! 5 7 1, 1 2 8 i u 1 v i 5 ! 9 1 49 4 5 ! 85 2 u 1 v 5 7 3, 7 2 8 i v i 5 ! 4 1 9 5 ! 13 i u i 5 ! 1 1 1 4 5 ! 5 2 u 5 7 1, 1 2 8 , v 5 k 2, 3 l 45. i u 1 v i i u i 1 i v i i u 1 v i 5 ! 74 ± 8.602 u 1 v 5 k 7, 5 l i v i 5 !
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ODD10 - CHAPTER 10 Vectors and the Geometry of Space...

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