Chapter 13

# For instance in example 7 the line l was given as the

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Unformatted text preview: tance, in Example 7 the line L was given as the line of intersection of the planes x y z 1 and x 2y 3z 1. The symmetric equations that we found for L could be written as z x =3 1 y 2 5 z 3 y 2 and _2 _1 y 0 1 2 _2 0 _1 x 1 which is again a pair of linear equations. They exhibit L as the line of intersection of the planes x 1 5 y 2 and y 2 z 3 . (See Figure 11.) In general, when we write the equations of a line in the symmetric form FIGURE 11 x |||| Figure 11 shows how the line L in Example 7 can also be regarded as the line of intersection of planes derived from its symmetric equations. x0 a y z y0 b z0 c we can regard the line as the line of intersection of the two planes x x0 y a y0 b y and y0 z z0 b c EXAMPLE 8 Find a formula for the distance D from a point P1 x 1, y1, z1 to the plane ax cz by d 0. SOLUTION Let P0 x 0 , y0 , z0 be any point in the given plane and let b be the vector corresponding to P0 P1. Then A b x 0 , y1 y0 , z1 z0 From Figure 12 you can see that the distance D from P1 to the plane is equal to the absolute value of the scalar projection of b onto the normal vector n a, b, c . (See Section 13.3.) Thus P¡ ¨ b x1 D n D nb n compn b b y1 y0 c z1 2 2 2 b c sa a x1 FIGURE 12 x0 a x1 P¸ b y1 c z1 sa 2 a x0 b y0 b c2 z0 c z0 2 Since P0 lies in the plane, its coordinates satisfy the equation of the plane and so we have ax 0 b y0 c z0 d 0. Thus, the formula for D can be written as D 9 a x1 b y1 c z1 d sa 2 b 2 c 2 EXAMPLE 9 Find the distance between the parallel planes 10 x 5x y z 2y 2z 5 and 1. SOLUTION First we note that the planes are parallel because their normal vectors 10, 2, 2 and 5, 1, 1 are parallel. To ﬁnd the distance D between the planes, we choose any point on one plane and calculate its distance to the other plane. In particular, if we put y z 0 in the equation of the ﬁrst plane, we get 10 x 5 and so 5E-13(pp 858-867) 1/18/06 11:30 AM Page 865 ❙❙❙❙ S ECTION 13.5 EQUATIONS OF LINES AND PLANES 865 ( 1 , 0, 0) is a point in this plane. By Formula 9,...
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