Chapter Review 677 In Problems 11–13, graph each polar equation. Be sure to test for symmetry. 11. θ = r 4 cos 12. θ = − r 3 3sin 13. θ = − r 4 cos In Problems 14 and 15, write each complex number in polar form and in exponential form. 14. − −i 1 15. − i 4 3 In Problems 16–18, write each complex number in rectangular form, and plot each in the complex plane. 16. π ⋅ e2 i 5 6 17. π π ( ) +i 3 cos 2 3 sin 2 3 18. π ⋅ e 0.1 i 35 18 In Problems 19–21, find zw and z w . Write your answers in polar form and in exponential form. 19. π π = + z i cos 4 9 sin 4 9 π π = + w i cos 5 18 sin 5 18 20. π π ( ) = + z i 3 cos 9 5 sin 9 5 π π ( ) = + w i 2 cos 5 sin 5 21. π π ( ) = + z i 5 cos 18 sin 18 π π = + w i cos 71 36 sin 71 36 In Problems 22–25, write each expression in exponential form, rectangular form, and polar form. 22. π π ( ) ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ i 3 cos 9 sin 9 3 23. π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ i 2 cos 5 8 sin 5 8 4 24. ( ) − i 1 3 6 25. ( ) + i 3 4 4 26. Find all the complex cube roots of 27. In Problems 27 and 28, use the figure to graph each of the following: 27. +u v 28. +u v 2 3 In Problems 29 and 30, the vector v is represented by the directed line segment PQ. Write v in the form + a b i j and find v . 29. ( ) ( ) = − = − P Q 1, 2; 3, 6 30. ( ) ( ) = − = − P Q 0, 2; 1, 1 In Problems 31–35, use the vectors = − + v i j 2 and = − w i j 4 3 to find: 31. +v w 32. −v w 4 3 33. v 34. + v w 35. A unit vector in the same direction as v. 36. Find the vector v in the xy-plane with magnitude 3 if the direction angle of v is ° 60 . 37. Find the direction angle α of = − + v i j3 . 38. Find the distance from ( ) = − P 1, 3, 2 1 to ( ) = − P 4, 2, 1 . 2 v u 39. A vector v has initial point ( ) = − P 1, 3, 2 and terminal point ( ) = − Q 4, 2, 1 . Write v in the form = + + a b c v i j k. In Problems 40–45, use the vectors = + − v i j k 3 2 and = − + − w i j k 3 2 to find each expression. 40. −v w 4 3 41. −v w 42. − v w 43. ×v w 44. ( ) ⋅ × v v w 45. Find a unit vector orthogonal to both v and w. In Problems 46–49, find the dot product ⋅ v w and the angle between v and w. 46. = − + = − v i j w i j 2 , 4 3 47. = − = − + v i j w i j 3 , 48. = + + = − + v i j k w i j k , 49. = − + = − − v i k w i j k 4 j 2 , 2 3 In Problems 50–52, determine whether v and w are parallel, orthogonal, or neither. 50. = + = − − v i j w i j 2 3 ; 4 6 51. = − + = − + v i j w i j 2 2 ; 3 2 52. = − = + v i j w i j 3 2 ; 4 6 In Problems 53 and 54, decompose v into two vectors, one parallel to w and the other orthogonal to w. 53. = + = − + v i w i j 2 j; 4 3 54. = + v i j 2 3 ; = + w i j 3 55. Find the direction angles of the vector = − + v i j k 3 4 2 . 56. Find the area of the parallelogram with vertices ( ) = P 1, 1, 1 , 1 ( ) ( ) = = P P 2, 3, 4 , 6, 5, 2 , 2 3 and ( ) = P 7, 7, 5 . 4 57. If × = − + u v i j k 2 3 , what is ×v u? 58. Suppose that = u v3 . What is ×u v?
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