Unit vector u u 2 3 5 38 2 38 3 38 5 38 2w v

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Unformatted text preview: x y 2z 3 (a) x 1, y 2 t, z 3 (b) None Solving simultaneously, we have z 1. Substituting z 1 into the second equation we have y x 1. Substituting for x in this equation we obtain two points on the line of intersection, 0, 1, 1 , 1, 0, 1 . The direction vector of the line of intersection is v i j. (a) x (b) x t, y y 1, z 1 1 t, z 1 Review Exercises for Chapter 10 259 47. The two lines are parallel as they have the same direction numbers, 2, 1, 1. Therefore, a vector parallel to the plane is v 2i j k. A point on the first line is 1, 0, 1 and a point on the second line is 1, 1, 2 . The vector u 2i j 3k connecting these two points is also parallel to the plane. Therefore, a normal to the plane is v u i 2 2 2i j 1 1 4j k 1 3 2i 1 x 2j . 2y 2y 0 1 49. Q 2x 1, 0, 2 3y \ 6z 6 A point P on the plane is 3, 0, 0 . PQ n D 2, \ 2, 0, 2 3, 6 8 7 PQ n n Equation of the plane: x 51. Q 3, 2, 4 point 53. x 2y 3z 6 P 5, 0, 0 point on plane n \ Plane Intercepts: 6, 0, 0 , 0, 3, 0 , 0, 0, 2 z 2, 5, 1 normal to plane 2, \ PQ D 2, 4 3 PQ n n 10 30 30 3 6 x (0, 0, 2) 3 (0, 3, 0) (6, 0, 0) y 55. y 1 z 2 57. x2 16 y2 9 z2 1 2 −4 z Plane with rulings parallel to the x-axis z 2 Ellipsoid xy-trace: xz-trace: x2 16 x2 16 y2 9 y2 9 z2 z2 1 1 1 4 x 5 −2 y 2 6 x y yz-trace: 59. x2 16 y2 9 z2 1 2 −2 5 z Hyperboloid of two sheets y2 xy-trace: 4 xz-trace: None yz-trace: y2 9 z2 1 x2 16 1 x 5 y 260 Chapter 10 x2 Vectors and the Geometry of Space y2 2 z 61. (a) rz 2z x2 y2 2z 2 0 1 2 4 3 −2 2 x 1 2 3 y 2 (b) V 2 0 2 x3 2x 0 12 x 2 13 x dx 2 x4 8 2 0 1 dx 3 y 2 2 4 2 2 x2 1 x 12.6 cm3 x3 12 2 1 2 3 (c) V 2 2 12 12 x 2 13 x dx 2 2 12 y 1 dx 3 2x x2 31 64 x4 8 2 1 2 4 x 1 2 3 225 64 11.04 cm 3 63. 2 2, 2 2, 2 , rectangular (a) r (b) 22 22 2 22 22 2 4, 2 2 arctan 2 5, 1 3 ,z 4 3 , 4 2, arccos 4, 3 , 2 , cylindrical 4 arccos 1 , 5 2 5, 3 5 , arccos , spherical 4 5 2 2 2 25 65. 100, 6 , 50 , cylindrical 502 50 5 67. 25, r2 3 , , spherical 44 25 sin 3 4 2 1002 6 arccos 50 5, ⇒r 25 2 2 50 50 5 arccos 1 5 4 63.4 z 25 cos 2 , 2 , 25 cos 3 4 25 2 2 6 , 63.4 , spherical 4 25 2 , cylindrical 2 69. x2 y2 2z r 2 sin2 cos2 2 (a) Cylindrical: r 2 cos2 (b) Spherical: 2 2z, r 2 cos 2 sin2 sin2 2z 2 cos , sin2 cos 2 2 cos 0, 2 sec 2 cos csc2 sin2...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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