Chapter 13

# For instance the planes x 2y 3z 4 and 2 x 4y 6z 3 are

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Unformatted text preview: el. For instance, the planes x 2y 3z 4 and 2 x 4y 6z 3 are parallel because their normal vectors are n1 1, 2, 3 and n 2 2, 4, 6 and n 2 2 n1 . If two planes are not parallel, then they intersect in a straight line and the angle between the two planes is deﬁned as the acute angle between their normal vectors (see angle in Figure 9). ¨ FIGURE 9 EXAMPLE 7 |||| Figure 10 shows the planes in Example 7 and their line of intersection L. x+y+z=1 x-2y+3z=1 (a) Find the angle between the planes x y z 1 and x 2y 3z 1. (b) Find symmetric equations for the line of intersection L of these two planes. SOLUTION (a) The normal vectors of these planes are 6 4 2 z0 _2 _4 n1 L and so, if 0 y 2 2 0 x n2 1, 2, 3 is the angle between the planes, Corollary 13.3.6 gives n1 n 2 n1 n 2 cos _2 1, 1, 1 _2 cos FIGURE 10 1 11 12 s1 1 1 s1 2 s42 13 49 2 s42 72 (b) We ﬁrst need to ﬁnd a point on L. For instance, we can ﬁnd the point where the line intersects the xy-plane by setting z 0 in the equations of both planes. This gives the equations x y 1 and x 2y 1, whose solution is x 1, y 0. So the point 1, 0, 0 lies on L. Now we observe that, since L lies in both planes, it is perpendicular to both of the normal vectors. Thus, a vector v parallel to L is given by the cross product |||| Another way to ﬁnd the line of intersection is to solve the equations of the planes for two of the variables in terms of the third, which can be taken as the parameter. v n1 i 1 1 n2 jk 11 23 5i 2j 3k and so the symmetric equations of L can be written as x 1 5 y 2 z 3 NOTE Since a linear equation in x, y, and z represents a plane and two nonparallel planes intersect in a line, it follows that two linear equations can represent a line. The points x, y, z that satisfy both a 1 x b1 y c1 z d1 0 and a 2 x b2 y c2 z d2 0 ■ 5E-13(pp 858-867) 864 ❙❙❙❙ 1/18/06 11:30 AM CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE L y x-1 = _2 5 2 1 z0 y _1 2 Page 864 lie on both of these planes, and so the pair of linear equations represents the line of intersection of the planes (if they are not parallel). For ins...
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