Chapter 13

# Equations x 1 y 5 15s z 72 find the distance between

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Unformatted text preview: . Show that the lines with symmetric equations x 58. (a) Find the point at which the given lines intersect: P1: 4 x t 10 z 3 t, ■ 2z 2y 56. Find an equation for the plane consisting of all points that are and 1 t, t 2, 2, equidistant from the points 1, 1, 0 and 0, 1, 1 . equidistant from the points z z 1 4 y t, ■ ■ |||| 68. 3 x 0 x t, 2, 7 , 4 x ■ 67. z 1 3y 5t Find the distance from the point to the given plane. |||| 67–68 2 ■ 65. 2, 8, 5 , 1 53–54 |||| Find parametric equations for the line of intersection of the planes. x x 1; ■ ■ 53. z 2 |||| Use the formula in Exercise 39 in Section 13.4 to ﬁnd the distance from the point to the given line. 1 ■ 51. x z t, y 63–64 2z y ■ L4: r 3 |||| ■ t, 1 (a) Find symmetric equations for the line of intersection of the planes and (b) ﬁnd the angle between the planes. ■ 1 1 3y 49. x 51–52 6y z y 48. 2 x ■ 3x identical? L1: x L3: x 1, 4y 47. x 62. Which of the following four lines are parallel? Are any of them L2: x 3z 4y 867 d a by 0 cz 0 0 74. Give a geometric description of each family of planes. (a) x y (b) x y (c) y cos zc cz 1 z sin 1 ■ 5E-13(pp 868-877) 868 ❙❙❙❙ 1/18/06 11:34 AM Page 868 CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE LABORATORY PROJECT Putting 3D in Perspective Computer graphics programmers face the same challenge as the great painters of the past: how to represent a three-dimensional scene as a ﬂat image on a two-dimensional plane (a screen or a canvas). To create the illusion of perspective, in which closer objects appear larger than those farther away, three-dimensional objects in the computer’s memory are projected onto a rectangular screen window from a viewpoint where the eye, or camera, is located. The viewing volume––the portion of space that will be visible––is the region contained by the four planes that pass through the viewpoint and edge of the screen window. If objects in the scene extend beyond these four planes, they must be truncated before pixel data are sent to the screen. These planes a...
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