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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 x2y+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 xyplane 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
■ 5E13(pp 858867) 864 ❙❙❙❙ 1/18/06 11:30 AM CHAPTER 13 VECTORS AND THE GEOMETRY OF SPACE L y
x1
= _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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 Fall '09
 hamrick

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