27 point 2 1 2 n 1x x 2 i 2 0 1 0 0 0y 1 0z 2 0 29

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Unformatted text preview: the lines: x x 41. Let v be the vector from 1, 1, 1 to 2, 2, 1 : v 3y z 3i 3: n j 2i 2k 3j k 1 y z y 5 5 z 1 Let n be a vector normal to the plane 2x Since v and n both lie in the plane p, the normal vector to p is v 7x 7x n 2 y i 3 2 j 1 3 1y 11z 5 k 2 1 2 7i 11 z j 1 11k 0 Section 10.5 Lines and Planes in Space 247 43. Let u i and let v be the vector from 1, 2, 5, 6 : v i 7j 7k 2, 1 to 45. The normal vectors to the planes are n1 Thus, 5, 3, 1 , n2 1, 4, 7 , cos n1 n2 n1 n2 0. Since u and v both lie in the plane P, the normal vector to P is: u y y z v 2 1 ijk 100 177 z 1 7j 0 7k 7j k 2 and the planes are orthogonal. 47. The normal vectors to the planes are n1 cos Therefore, i 3j 6k, n2 5 5i 3 j 6 k, 4 138 . 414 1, 5, 1 49. The normal vectors to the planes are n1 and n2 5, 25, 5 . Since n2 5n1, the planes are parallel, but not equal. n1 n2 n1 n2 arccos 46 27 4 138 414 83.5 . 51. 4x 2y 6z z 6 4 12 53. 2x y 3z z 3 2 4 55. y z 5 z 6 −4 6 x 4 6 y x 3 −1 y x 6 6 y 57. x 5 z 3 59. 2x y z z 2 6 61. 5x −2 4y 6z z 8 0 6 x 4 2 4 6 y 5 x 5 y −6 Generated by Maple x 2 −1 1 y Generated by Maple 63. P1: n P2: n P3: n P4: n 3, 2, 5 6, 4, 3, 2, 5 10 1, 1, 1, 1 on plane 1, 1 not on plane 65. Each plane passes through the points c, 0, 0 , 0, c, 0 , and 0, 0, c . 75, 50, 125 1, 1, 1 on plane P1 and P4 are identical. P1 P4 is parallel to P2. 248 Chapter 10 Vectors and the Geometry of Space 69. Writing the equation of the line in parametric form and substituting into the equation of the plane we have: x 2 1 2 1 2 t t, y 2 3 2 3 2 t t, z 1 1 2t 2t 12, t 3 2 67. The normals to the planes are n1 3i 2j k and n2 i 4j 2k. The direction vector for the line is n2 n1 i 1 3 2y 2z x j 4 2 14 0 14 2 k 2 1 7j 2k . Now find a point of intersection of the planes. 6x x 7x 4y 4y Substituting t 3 2 into the parametric equations for the line we have the point of intersection 2, 3, 2 . The line does not lie in the plane. Substituting 2 for x in the second equation, we have 4y 2z 2 or z 2y 1. Letting y 1, a point of intersection is 2, 1, 1 . x 2, y...
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