Unformatted text preview: 62 CHAPTER 1. COORDINATES AND VECTORS which we can calculate as
→→→
−−−
N · ( q − p 0)
=
→
−
N
→→ →→
−− −−
N · q −N · p0
=
→→
−−
N ·N
(Ax + By + Cz ) − D 
√
=
.
A2 + B 2 + C 2
For example, the distance from Q(1, 1, 2) to the plane P given by
2x − 3y + z = 5
is
dist(Q, P ) = (2)(1) − 3(1) + 1(2) − (5) 4
=√ .
14 22 + (−3)2 + 12 The distance between two parallel planes is the distance from any
point Q on one of the planes to the other plane. Thus, the distance
between the parallel planes discussed earlier
4x −6y +2z = 8
−6x +9y −3z = 12
is the same as the distance from Q(3, −1, −5), which lies on the ﬁrst plane,
to the second plane, or
dist(P1 , P2 ) = dist(Q, P2 )
(−6)(3) + (9)(−1) + (−3)(−5) − (12)
=
(−6)2 + (9)2 + (−3)2
24
=√.
3 14
Finally, the angle θ between two planes P1 and P2 can be deﬁned as
follows (Figure 1.33): if they are parallel, the angle is zero. Otherwise,
they intersect along a line ℓ0 : pick a point P0 on ℓ0 , and consider the line ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Vectors

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