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Unformatted text preview: 2, b 3, c 4, x 0
see that an equation of the plane is (0, 0, 3) (6, 0, 0) 4, and z0 2, y0 1 in Equation 7, we 2 3y or 4 4z 1 0 2x 2x
(0, 4, 0) 1 with normal vector 2, 3, 4 . Find the intercepts and sketch the plane. n 3y 4z 12 y To ﬁnd the xintercept we set y z 0 in this equation and obtain x 6. Similarly, the
yintercept is 4 and the zintercept is 3. This enables us to sketch the portion of the plane
that lies in the ﬁrst octant (see Figure 7). x FIGURE 7 By collecting terms in Equation 7 as we did in Example 4, we can rewrite the equation
of a plane as
ax 8 by cz d 0 where d
a x 0 b y0 cz0 . Equation 8 is called a linear equation in x, y, and z. Conversely, it can be shown that if a, b, and c are not all 0, then the linear equation (8) represents a plane with normal vector a, b, c . (See Exercise 73.)
 Figure 8 shows the portion of the plane in
Example 5 that is enclosed by triangle PQR. EXAMPLE 5 Find an equation of the plane that passes through the points P 1, 3, 2 , Q 3, 1, 6 , and R 5, 2, 0 .
l Q(3, 1, 6) a
P(1, 3, 2) 2, 4, 4 b 4, 1, Since both a and b lie in the plane, their cross product a
and can be taken as the normal vector. Thus y n x a i
2
4 b R(5, 2, 0)
FIGURE 8 l SOLUTION The vectors a and b corresponding to PQ and PR are z j
4
1 k
4
2 2 b is orthogonal to the plane 12 i 20 j 14 k With the point P 1, 3, 2 and the normal vector n, an equation of the plane is
12 x
or 1 20 y 3 14 z 2 0 6x 10 y 7z 50 5E13(pp 858867) 1/18/06 11:29 AM Page 863 SECTION 13.5 EQUATIONS OF LINES AND PLANES E XAMPLE 6 Find the point at which the line with parametric equations x y 4 t, z 5 t intersects the plane 4 x 2z 5y 2 ❙❙❙❙ 863 3 t, 18. SOLUTION We substitute the expressions for x, y, and z from the parametric equations into
the equation of the plane: 42 3t 5 4t 25 t 18 This simpliﬁes to 10 t 20, so t
2. Therefore, the point of intersection occurs
when the parameter value is t
2. Then x 2 3 2
4, y
42
8,
z 5 2 3 and so the point of intersection is 4, 8, 3 .
n™ ¨ n¡ Two planes are parallel if their normal vectors are parall...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
 Fall '09
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