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Unformatted text preview: (8/1/08) Math 20C. Lecture Examples. Section 12.2. Vectors in three dimensions To use rectangular xyz-coordinates in three-dimensional space, we introduce mutually perpendicular x-, y-, and z-axes intersecting at their origins, as in Figure 1. These axes form xy-, xz-, and yz-coordinate planes, which divide the space into eight octants . FIGURE 1 FIGURE 2 The coordinates ( x,y,z ) of a point in space are determined by planes passing through the point and perpendicular to the coordinate axes (Figure 2). Example 1 Sketch the box consisting of all points ( x,y,z ) with 0 x 2 , 2 y 3, and 0 z 2. What are the coordinates of the eight corners of the box? Answer: Figure A1. The corners of its base, ordered counterclockwise, are (2 , 2 , 0) , (2 , 3 , 0) , (0 , 3 , 0), and (0 , 2 , 0). The corners of its top are (2 , 2 , 2) , (2 , 3 , 2) , (0 , 3 , 2), and (0 , 2 , 2). Figure A1 Lecture notes to accompany Section 12.2 of Calculus, Early Transcendentals by Rogawski. 1 Math 20C. Lecture Examples. (8/1/08) Section 12.2, p. 2 The Pythagorean Theorem and the distance between two points If a rectangular box has length a , width b , and height c , as in Figure 6, then, by the Pythagorean Theorem for a right triangle, the length of a diagonal of its base is a 2 + b 2 . Then, because the diagonal of the box is the hypotenuse of a right triangle with base of length a 2 + b 2 and height c (Figure 7), its length is the square root of [ a 2 + b 2...
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