Chapter 13

# If neither nd the angle between them 62 which of the

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Unformatted text preview: planes are parallel, perpendicular, or neither. If neither, ﬁnd the angle between them. 45. x 46. 2 z 7z x, 3x 12 y 6z 50. x y z 5, 2 z, x 6y 1 2x ■ 0 4z 4z 8y 2z 2y ■ x 1, 4z 4y 1 y 2 1 t, y 2, 1, 3 1, 2 x ■ ■ ■ y z 2, 3x 4y 5z 52. x 2y z 1, 2x y z ■ ■ ■ ■ ■ ■ ■ ■ 63. 1, 2, 3 ; 64. 1, 0, ■ 65–66 6 ■ ■ ■ ■ ■ ■ 54. 2 x ■ y, 5z ■ 2x 3 ■ z 5y 0, ■ x 66. 3, ■ ■ z 2 ■ ■ ■ ■ ■ ■ 55. Find an equation for the plane consisting of all points that are ■ 5 2 y ■ x 4, 2, 1 and 2, 4, 3 . 57. Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c. r 1, 1, 0 t 1, r 2, 0, 2 s (b) Find an equation of the plane that contains these lines. 59. Find parametric equations for the line through the point z 2 and t, z 2 t. 60. Find parametric equations for the line through the point 0, 1, 2 that is perpendicular to the line x y 1 t, z 2 t and intersects this line. x 2y z ■ 9z 6y ■ 1, ■ P3: 6x 3y 9z P2: 4 x 3 5 P4: z 2z 2y 2x ■ ■ ■ ■ ■ ■ ■ 4, 3z 6y x 3z 2y ■ ■ cz by d1 1 ■ ■ 0 and ax ■ sa 2 cz by d1 D ■ ■ 0 is d2 d2 b 2 c2 70. Find equations of the planes that are parallel to the plane x y 6 3 ■ 4 69. Show that the distance between the parallel planes ax ■ 5 ■ 3x ■ 2z 2y 1 and two units away from it. y z and z 3 are skew, and ﬁnd the distance between equations x 1 y 5 15s, z t, y 2 1 6 s. 6 t, z 2 t, and x 1 2 s, 73. If a, b, and c are not all 0, show that the equation a x b y c z d 0 represents a plane and a, b, c is a normal vector to the plane. Hint: Suppose a 0 and rewrite the equation in the form ax them identical? 6z ■ Find the distance between the given parallel planes. t, 1 61. Which of the following four planes are parallel? Are any of 2y 2t ■ 72. Find the distance between the skew lines with parametric 1, 1, 0 y 1 1 ■ 1 6y ■ x 1 y2 these lines. 1, 2 0, 1, 2 that is parallel to the plane x perpendicular to the line x 1 t, y z 3 t, 5t 71...
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