SECTION 9.6 Vectors in Space 667 ‘Are You Prepared?’ The answer is given at the end of these exercises. If you get the wrong answer, read the page listed in red. 9.6 Assess Your Understanding 1. The distance d from P x y , 1 1 1 ( ) = to P x y , 2 2 2 ( ) = is d = . (p. 14) Skill Building In Problems 7–14, describe the set of points x y z , , ( ) defined by the equation(s). 7. y 0 = 8. x 0 = 9. z 2 = 10. y 3 = 11. x 4 = − 12. z 3 = − 13. x 1 = and y 2 = 14. x 3 = and z 1 = Concepts and Vocabulary 2. If a b c v i j k = + + is a vector in space, the scalars a , b , c are called the of v . 3. The squares of the direction cosines of a vector in space add up to . 4. True or False In space, the dot product of two vectors is a positive number. 5. True or False A vector in space may be described by specifying its magnitude and its direction angles. 6. Multiple Choice In space, points of the form x y , , 0 ( ) lie in: (a) the xy -plane (b) the xz -plane (c) the yz -plane (d) none of these In Problems 15–20, find the distance from P1 to P .2 15. P 0, 0, 0 1 ( ) = and P 4, 1, 2 2 ( ) = 16. P 0, 0, 0 1 ( ) = and P 1, 2, 3 2 ( ) = − 17. P 1, 2, 3 1 ( ) = − − and P 0, 2, 1 2 ( ) = − 18. P 2, 2, 3 1 ( ) = − and P 4, 0, 3 2 ( ) = − 19. P 4, 2, 2 1 ( ) = − − and P 3, 2, 1 2 ( ) = 20. P 2, 3, 3 1 ( ) = − − and P 4, 1, 1 2 ( ) = − In Problems 21–26, opposite vertices of a rectangular box whose edges are parallel to the coordinate axes are given. List the coordinates of the other six vertices of the box. 21. 0, 0, 0 ; 2, 1, 3 ( ) ( ) 22. 0, 0, 0 ; 4, 2, 2 ( ) ( ) 23. 1, 2, 3 ; 3, 4, 5 ( ) ( ) 24. 5, 6, 1 ; 3, 8, 2 ( ) ( ) 25. 1, 0, 2 ; 4, 2, 5 ( ) ( ) − 26. 2, 3, 0 ; 6, 7, 1 ( ) ( ) − − − In Problems 27–32, the vector v has initial point P and terminal point Q. Write v in the form a b c i j k; + + that is, find its position vector. 27. P Q 0, 0, 0 ; 3, 4, 1 ( ) ( ) = = − 28. P Q 0, 0, 0 ; 3, 5, 4 ( ) ( ) = = − − 29. P Q 3, 2, 1 ; 5, 6, 0 ( ) ( ) = − = 30. P Q 3, 2, 0 ; 6, 5, 1 ( ) ( ) = − = − 31. P Q 2, 1, 4 ; 6, 2, 4 ( ) ( ) = − − = − 32. P Q 1, 4, 2 ; 6, 2, 2 ( ) ( ) = − − = In Problems 33–38, find v . 33. v i j k 3 6 2 = − − 34. v i j k 6 12 4 = − + + 35. v i j k = − + 36. v i j k = − − + 37. v i j k 2 3 3 = − + − 38. v i j k 6 2 2 = + − In Problems 39–44, find each quantity if v i j k 3 5 2 = − + and w i j k 2 3 2 . = − + − 39. v w 2 3 + 40. v w 3 2 − 41. v w− 42. v w + 43. v w − 44. v w + In Problems 45–50, find the unit vector in the same direction as v . 45. v i5 = 46. v j3 = − 47. v i j k 3 6 2 = − − 48. v i j k 6 12 4 = − + + 49. v i j k = + + 50. v i j k 2 = − + In Problems 51–58, find the dot product v w⋅ and the angle between v and w . 51. v i j w i j k , = − = + + 52. v i j w i j k , = + = − + − 53. v i j k w i j k 2 3 , 2 2 = + − = + + 54. v i j k w i j k 2 2 , 2 3 = + − = + + 55. v i j k w i j k 3 2 , = − + = + − 56. v i j k w i j k 3 2 , = + + = − + 57. v i j k w i j k 3 4 , 6 8 2 = + + = + + 58. v i j k w i j k 3 4 , 6 8 2 = − + = − + 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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