MATH
ISM_T11_C12_A

# ISM_T11_C12_A - CHAPTER 12 VECTORS AND THE GEOMETRY OF...

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE 12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS 1. The line through the point ( ) parallel to the z-axis #ß \$ß ! 2. The line through the point ( 1 0 ) parallel to the y-axis ß ß ! 3. The x-axis 4. The line through the point (1 ) parallel to the z-axis ß !ß ! 5. The circle x y 4 in the xy-plane # # œ 6. The circle x y 4 in the plane z = 2 # # œ 7. The circle x z 4 in the xz-plane # # œ 8. The circle y z 1 in the yz-plane # # œ 9. The circle y z 1 in the yz-plane # # œ 10. The circle x z 9 in the plane y 4 # # œ œ 11. The circle x y 16 in the xy-plane # # œ 12. The circle x z 3 in the xz-plane # # œ 13. (a) The first quadrant of the xy-plane (b) The fourth quadrant of the xy-plane 14. (a) The slab bounded by the planes x 0 and x 1 œ œ (b) The square column bounded by the planes x 0, x 1, y 0, y 1 œ œ œ œ (c) The unit cube in the first octant having one vertex at the origin 15. (a) The solid ball of radius 1 centered at the origin (b) The exterior of the sphere of radius 1 centered at the origin 16. (a) The circumference and interior of the circle x y 1 in the xy-plane # # œ (b) The circumference and interior of the circle x y 1 in the plane z 3 # # œ œ (c) A solid cylindrical column of radius 1 whose axis is the z-axis 17. (a) The closed upper hemisphere of radius 1 centered at the origin (b) The solid upper hemisphere of radius 1 centered at the origin 18. (a) The line y x in the xy-plane œ (b) The plane y x consisting of all points of the form (x x z) œ ß ß

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780 Chapter 12 Vectors and the Geometry of Space 19. (a) x 3 (b) y 1 (c) z 2 œ œ œ 20. (a) x 3 (b) y 1 (c) z 2 œ œ œ 21. (a) z 1 (b) x 3 (c) y 1 œ œ œ 22. (a) x y 4, z 0 (b) y z 4, x 0 (c) x z 4, y 0 # # # # # # œ œ œ œ œ œ 23. (a) x (y 2) 4, z 0 (b) (y 2) z 4, x 0 (c) x z 4, y 2 # # # # # # œ œ œ œ œ œ 24. (a) (x 3) (y 4) 1, z 1 (b) (y 4) (z 1) 1, x 3 œ œ œ œ # # # # (c) (x 3) (z 1) 1, y 4 œ œ # # 25. (a) y 3, z 1 (b) x 1, z 1 (c) x 1, y 3 œ œ œ œ œ œ 26. x y z x (y 2) z x y z x (y 2) z y y 4y 4 y 1 È È # # # # # # # # # # # # # # œ Ê œ Ê œ Ê œ 27. x y z 25, z 3 x y 16 in the plane z 3 # # # # # œ œ Ê œ œ 28. x y (z 1) 4 and x y (z 1) 4 x y (z 1) x y (z 1) z 0, x y 3 # # # # # # # # # # # # # # œ œ Ê œ Ê œ œ 29. 0 z 1 30. 0 x 2, 0 y 2, 0 z 2 Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ 31. z 0 32. z 1 x y Ÿ œ È # # 33. (a) (x 1) (y 1) (z 1) 1 (b) (x 1) (y 1) (z 1) 1 # # # # # # 34. 1 x y z 4 Ÿ Ÿ # # # 35. P P 3 1 3 1 0 1 9 3 k k a b a b a b É È " # # # # œ œ œ 36. P P 2 1 5 1 0 5 50 5 2 k k a b a b a b É È È " # # # # œ œ œ 37. P P 4 1 2 4 7 5 49 7 k k a b a b a b É È " # # # # œ œ œ 38. P P 2 3 3 4 4 5 3 k k a b a b a b É È " # # # # œ œ 39. P P 2 0 2 0 2 0 3 4 2 3 k k a b a b a b É È È " # # # # œ œ œ 40. P P 0 5 0 3 0 2 38 k k a b a b a b É È " # # # # œ œ 41. center ( 2 0 2), radius 2 2 42. center , radius ß ß ß ß È ˆ " " " # # # # È 21 43. center 2 2 2 , radius 2 44. center , radius Š È È È È ˆ ß ß ß " " 3 3 3 29 È
Section 12.2 Vectors 781 45. (x 1) (y 2) ( 3) 14 46. x (y 1) (z 5) 4 D œ œ # # # # # # 47. (x 2) y z 3 48. x (y 7) z 49 œ œ # # # # # # 49. x y z 4x 4z 0 x 4x 4 y z 4z 4 4 4 # # # # # # œ Ê œ a b a b (x 2) (y 0) (z 2) 8 the center is at ( 2 0 2) and the radius is 8 Ê œ Ê ß ß # # # # Š È È 50. x y z 6y 8z 0 x y 6y 9 z 8z 16 9 16 (x 0) (y 3) (z 4) 5 # # # # # # # # # # œ Ê œ Ê œ a b a b the center is at (0 3 4) and the radius is 5 Ê ß ß 51. 2x 2y 2z x y z 9 x x y y z z # # # # # # " " " # # # # œ Ê œ 9 x x y y z z x y z Ê œ Ê œ ˆ ˆ

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• Spring '07
• BERMAN
• Calculus, Vector Motors, e3, C5-convertase, Geometry of Space

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