James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

832 Chapter 12 Vectors and the Geometry of Space
34. The plane through the points s3, 0, 21d, s22, 22, 3d, and
s7, 1, 24d
35. The plane that passes through the point s3, 5, 21d and contains
the line x − 4 2 t, y − 2t 2 1, z − 23t
36. The plane that passes through the point s6, 21, 3d and
contains the line with symmetric equations
xy3 − y 1 4 − zy2
37. The plane that passes through the point s3, 1, 4d and contains
the line of intersection of the planes x 1 2y 1 3z − 1 and
2x 2 y 1 z − 23
38. The plane that passes through the points s0, 22, 5d and
s21, 3, 1d and is perpendicular to the plane 2z − 5x 1 4y
39. The plane that passes through the point s1, 5, 1d and is perpendicular
to the planes 2x 1 y 2 2z − 2 and x 1 3z − 4
40. The plane that passes through the line of intersection of the
planes x 2 z − 1 and y 1 2z − 3 and is perpendicular to the
plane x 1 y 2 2z − 1
41–44 Use intercepts to help sketch the plane.
41. 2x 1 5y 1 z − 10 42. 3x 1 y 1 2z − 6
43. 6x 2 3y 1 4z − 6 44. 6x 1 5y 2 3z − 15
45–47 Find the point at which the line intersects the given plane.
45. x − 2 2 2t, y − 3t, z − 1 1 t; x 1 2y 2 z − 7
46. x − t 2 1, y − 1 1 2t, z − 3 2 t; 3x 2 y 1 2z − 5
47. 5x − yy2 − z 1 2; 10x 2 7y 1 3z 1 24 − 0
48. Where does the line through s23, 1, 0d and s21, 5, 6d intersect
the plane 2x 1 y 2 z − 22?
49. Find direction numbers for the line of intersection of the planes
x 1 y 1 z − 1 and x 1 z − 0.
50. Find the cosine of the angle between the planes x 1 y 1 z − 0
and x 1 2y 1 3z − 1.
51–56 Determine whether the planes are parallel, perpendicular,
or neither. If neither, find the angle between them. (Round to one
decimal place.)
51. x 1 4y 2 3z − 1, 23x 1 6y 1 7z − 0
52. 9x 2 3y 1 6z − 2, 2y − 6x 1 4z
53. x 1 2y 2 z − 2, 2x 2 2y 1 z − 1
54. x 2 y 1 3z − 1, 3x 1 y 2 z − 2
55. 2x 2 3y − z, 4x − 3 1 6y 1 2z
56. 5x 1 2y 1 3z − 2, y − 4x 2 6z
57–58 (a) Find parametric equations for the line of intersection of
the planes and (b) find the angle between the planes.
57. x 1 y 1 z − 1, x 1 2y 1 2z − 1
58. 3x 2 2y 1 z − 1, 2x 1 y 2 3z − 3
59–60 Find symmetric equations for the line of intersection of the
planes.
59. 5x 2 2y 2 2z − 1, 4x 1 y 1 z − 6
60. z − 2x 2 y 2 5, z − 4x 1 3y 2 5
61. Find an equation for the plane consisting of all points that are
equidistant from the points s1, 0, 22d and s3, 4, 0d.
62. Find an equation for the plane consisting of all points that are
equidistant from the points s2, 5, 5d and s26, 3, 1d.
63. Find an equation of the plane with x-intercept a, y-intercept b,
and z-intercept c.
64. (a) Find the point at which the given lines intersect:
r − k1, 1, 0 l 1 tk1, 21, 2 l
r − k2, 0, 2 l 1 sk21, 1, 0 l
(b) Find an equation of the plane that contains these lines.
65. Find parametric equations for the line through the point
s0, 1, 2d that is parallel to the plane x 1 y 1 z − 2 and
perpendicular to the line x − 1 1 t, y − 1 2 t, z − 2t.
66. Find parametric equations for the line through the point
s0, 1, 2d that is perpendicular to the line x − 1 1 t,
y − 1 2 t, z − 2t and intersects this line.
67. Which of the following four planes are parallel? Are any of
them identical?
P 1: 3x 1 6y 2 3z − 6 P 2: 4x 2 12y 1 8z − 5
P 3: 9y − 1 1 3x 1 6z P 4: z − x 1 2y 2 2
68. Which of the following four lines are parallel? Are any of them
identical?
L 1: x − 1 1 6t, y − 1 2 3t, z − 12t 1 5
L 2: x − 1 1 2t, y − t, z − 1 1 4t
L 3: 2x 2 2 − 4 2 4y − z 1 1
L 4: r − k3, 1, 5 l 1 tk4, 2, 8 l
69–70 Use the formula in Exercise 12.4.45 to find the distance
from the point to the given line.
69. s4, 1, 22d; x − 1 1 t, y − 3 2 2t, z − 4 2 3t
70. s0, 1, 3d; x − 2t, y − 6 2 2t, z − 3 1 t
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