AWL/Thomas_ch12p848-905 4100 8/25/04 2:44 PM Page 887
12.5 Lines and Planes in Space
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EXERCISES 12.5
Lines and Line Segments
Find parametric equations for the lines in Exercises 112. 1. The line through the point Ps 3, - 4, - 1 d parallel to the vector i+j+k 2. The line through Ps 1, 2, - 1 d and Qs - 1, 0, 1 d 3. The line through Ps - 2, 0, 3 d and Qs 3, 5, - 2 d 4. The line through P(1, 2, 0) and Qs 1, 1, - 1 d 5. The line through the origin parallel to the vector 2j + k 6. The line through the point s 3, - 2, 1 d parallel to the line x = 1 + 2t, y = 2 - t, z = 3t 7. The line through (1, 1, 1) parallel to the z-axis 8. The line through (2, 4, 5) perpendicular to the plane 3x + 7y - 5z = 21 9. The line through s 0, - 7, 0 d perpendicular to the plane x + 2y + 2z = 13
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Chapter 12: Vectors and the Geometry of Space 36. s 2, 1, - 1 d; 37. s 3, - 1, 4 d; 38. s - 1, 4, 3 d; x = 2t, y = 1 + 2t, y = - 3, z = 2t z = - 5 + 3t z = 4t
10. The line through (2, 3, 0) perpendicular to the vectors u = i + 2j + 3k and v = 3i + 4j + 5k 11. The x-axis 12. The z-axis
x = 4 - t, x = 10 + 4t,
y = 3 + 2t,
Find parametrizations for the line segments joining the points in Exercises 1320. Draw coordinate axes and sketch each segment, indicating the direction of increasing t for your parametrization. 13. (0, 0, 0), 15. (1, 0, 0), 17. s 0, 1, 1 d, 19. (2, 0, 2), (1, 1, 0) (1, 1, 3> 2) 14. (0, 0, 0), 16. (1, 1, 0), 18. (0, 2, 0), 20. s 1, 0, - 1 d, (1, 0, 0) (1, 1, 1) (3, 0, 0) s 0, 3, 0 d
In Exercises 3944, find the distance from the point to the plane. 39. s 2, - 3, 4 d, 40. s 0, 0, 0 d, 41. s 0, 1, 1 d, 42. s 2, 2, 3 d, 43. s 0, - 1, 0 d, 44. s 1, 0, - 1 d, x + 2y + 2z = 13 3x + 2y + 6z = 6 4y + 3z = - 12 2x + y + 2z = 4 2x + y + 2z = 4 - 4x + y + z = 4
s 0, - 1, 1 d (0, 2, 0)
Planes
Find equations for the planes in Exercises 2126. 21. The plane through P0s 0, 2, - 1 d normal to n = 3i - 2j - k 22. The plane through s 1, - 1, 3 d parallel to the plane 3x + y + z = 7 23. The plane through s 1, 1, - 1 d, s 2, 0, 2 d , and s 0, - 2, 1 d 24. The plane through (2, 4, 5), (1, 5, 7), and s - 1, 6, 8 d 25. The plane through P0s 2, 4, 5 d perpendicular to the line x = 5 + t, y = 1 + 3t, z = 4t
45. Find the distance from the plane x + 2y + 6z = 1 to the plane x + 2y + 6z = 10 . 46. Find the distance from the line x = 2 + t, y = 1 + t, z = - s 1> 2 d - s 1> 2 dt to the plane x + 2y + 6z = 10 .
Angles
Find the angles between the planes in Exercises 47 and 48. 47. x + y = 1, 2x + y - 2z = 2 x - 2y + 3z = - 1 48. 5x + y - z = 10,
26. The plane through As 1, - 2, 1 d perpendicular to the vector from the origin to A 27. Find the point of intersection of the lines x = 2t + 1, y = 3t + 2, z = 4t + 3 , and x = s + 2, y = 2s + 4, z = - 4s - 1 , and then find the plane determined by these lines. 28. Find the point of intersection of the lines x = t, y = - t + 2, z = t + 1 , and x = 2s + 2, y = s + 3, z = 5s + 6 , and then find the plane determined by these lines. In Exercises 29 and 30, find the plane determined by the intersecting lines. 29. L1: x = - 1 + t, y = 2 + t, z = 1 - t; - q 6 t 6 q L2: x = 1 - 4s, y = 1 + 2s, z = 2 - 2s; - q 6 s 6 q 30. L1: x = t, y = 3 - 3t, z = - 2 - t; - q 6 t 6 q L2: x = 1 + s, y = 4 + s, z = - 1 + s; - q 6 s 6 q 31. Find a plane through P0s 2, 1, - 1 d and perpendicular to the line of intersection of the planes 2x + y - z = 3, x + 2y + z = 2 . 32. Find a plane through the points P1s 1, 2, 3 d, P2s 3, 2, 1 d and perpendicular to the plane 4x - y + 2z = 7 .
T Use a calculator to find the acute angles between the planes in Exercises 4952 to the nearest hundredth of a radian. 49. 2x + 2y + 2z = 3, 50. x + y + z = 1, 51. 2x + 2y - z = 3, 52. 4y + 3z = - 12, 2x - 2y - z = 5 s the xy-plane d x + 2y + z = 2 3x + 2y + 6z = 6 z=0
Intersecting Lines and Planes
In Exercises 5356, find the point in which the line meets the plane. 53. x = 1 - t, 54. x = 2, 55. x = 1 + 2t, 56. x = - 1 + 3t, y = 3t, z = 1 + t; z = - 2 - 2t; z = 3t; z = 5t; 2x - y + 3z = 6 6x + 3y - 4z = - 12 x+y+z=2 2x - 3z = 7 y = 3 + 2t,
y = 1 + 5t, y = - 2,
Find parametrizations for the lines in which the planes in Exercises 5760 intersect. x 57. + y + z = 1, 59. x - 2y + 4z = 2, x+y=2 2x + y - 2z = 2 x + y - 2z = 5 58. 3x - 6y - 2z = 3, 60. 5x - 2y = 11,
Distances
In Exercises 3338, find the distance from the point to the line. 33. s 0, 0, 12 d; 34. s 0, 0, 0 d; 35. s 2, 1, 3 d; x = 4t, y = - 2t, z = 2t z = - 3 - 5t z=3 x = 5 + 3t, x = 2 + 2t, y = 5 + 4t, y = 1 + 6t,
4y - 5z = - 17
Given two lines in space, either they are parallel, or they intersect, or they are skew (imagine, for example, the flight paths of two planes in the sky). Exercises 61 and 62 each give three lines. In each exercise, determine whether the lines, taken two at a time, are parallel, intersect, or are skew. If they intersect, find the point of intersection.
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12.5 Lines and Planes in Space 61. L1: x = 3 + 2t, y = - 1 + 4t, z = 2 - t; - q 6 t 6 q L2: x = 1 + 4s, y = 1 + 2s, z = - 3 + 4s; - q 6 s 6 q L3: x = 3 + 2r, y = 2 + r, z = - 2 + 2r; - q 6 r 6 q 62. L1: x = 1 + 2t, L2: x = 2 - s, L3: x = 5 + 2r, y = - 1 - t, y = 3s, y = 1 - r, z = 1 + s; -q 6 t 6 q -q 6 s 6 q z = 8 + 3r ; - q 6 r 6 q z = 3t;
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72. Suppose L1 and L2 are disjoint (nonintersecting) nonparallel lines. Is it possible for a nonzero vector to be perpendicular to both L1 and L2 ? Give reasons for your answer.
Computer Graphics
73. Perspective in computer graphics In computer graphics and perspective drawing, we need to represent objects seen by the eye in space as images on a two-dimensional plane. Suppose that the eye is at Es x0, 0, 0 d as shown here and that we want to represent a point P1s x1, y1, z1 d as a point on the yz-plane. We do this by projecting P1 onto the plane with a ray from E. The point P1 will be portrayed as the point P(0, y, z). The problem for us as graphics designers is to find y and z given E and P1 . 1 1 a. Write a vector equation that holds between EP and EP1 . Use the equation to express y and z in terms of x0 , x1, y1 , and z1 . b. Test the formulas obtained for y and z in part (a) by investigating their behavior at x1 = 0 and x1 = x0 and by seeing what happens as x0 : q . What do you find?
z
Theory and Examples
63. Use Equations (3) to generate a parametrization of the line through Ps 2, - 4, 7 d parallel to v1 = 2i - j + 3k . Then generate another parametrization of the line using the point P2s - 2, - 2, 1 d and the vector v2 = - i + s 1> 2 d j - s 3> 2 d k .
64. Use the component form to generate an equation for the plane through P1s 4, 1, 5 d normal to n1 = i - 2j + k . Then generate another equation for the same plane using the point P2s 3, - 2, 0 d and the normal vector n2 = - 22i + 2 22j - 22k . 65. Find the points in which the line x = 1 + 2t, y = - 1 - t, z = 3t meets the coordinate planes. Describe the reasoning behind your answer. 66. Find equations for the line in the plane z = 3 that makes an angle of p> 6 rad with i and an angle of p> 3 rad with j. Describe the reasoning behind your answer. 67. Is the line x = 1 - 2t, y = 2 + 5t, z = - 3t parallel to the plane 2x + y - z = 8 ? Give reasons for your answer. 68. How can you tell when two planes A1 x + B1 y + C1 z = D1 and A2 x + B2 y + C2 z = D2 are parallel? Perpendicular? Give reasons for your answer. 69. Find two different planes whose intersection is the line x = 1 + t, y = 2 - t, z = 3 + 2t . Write equations for each plane in the form Ax + By + Cz = D . 70. Find a plane through the origin that meets the plane M: 2x + 3y + z = 12 in a right angle. How do you know that your plane is perpendicular to M?
P(0, y, z) P1(x1, y1, z1) 0 (x1, y1, 0) y
E (x 0, 0, 0) x
71. For any nonzero numbers a, b, and c, the graph of s x> a d + s y> b d + s z> c d = 1 is a plane. Which planes have an equation of this form?
74. Hidden lines Here is another typical problem in computer graphics. Your eye is at (4, 0, 0). You are looking at a triangular plate whose vertices are at (1, 0, 1), (1, 1, 0), and s - 2, 2, 2 d . The line segment from (1, 0, 0) to (0, 2, 2) passes through the plate. What portion of the line segment is hidden from your view by the plate? (This is an exercise in finding intersections of lines and planes.)
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