Engineering Calculus Notes 74

Engineering Calculus Notes 74 - 62 CHAPTER 1. COORDINATES...

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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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