Engineering Calculus Notes 76

Engineering Calculus Notes 76 - 64 CHAPTER 1. COORDINATES...

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Unformatted text preview: 64 CHAPTER 1. COORDINATES AND VECTORS ℓi in Pi (i = 1, 2) through P0 and perpendicular to ℓ0 . Then θ is by definition the angle between ℓ1 and ℓ2 . To relate this to the equations of P1 and P2 , consider the plane P0 (through P0 ) containing the lines ℓ1 and ℓ2 . P0 is perpendicular to ℓ0 and hence contains the arrows with tails at P0 representing the normals → − → − → − → − → → → → N 1 = A1 − + B1 − + C1 k (resp. N 2 = A2 − + B2 − + C2 k ) to P1 (resp. ı ı → − P2 ). But since N i is perpendicular to ℓi for i = 1, 2, the angle between the → − → − vectors N 1 and N 2 is the same as the angle between ℓ1 and ℓ2 (Figure 1.34), hence →→ −− N1 · N2 cos θ = − − (1.23) →→ N1 N2 For example, the planes determined by the two equations x + √ y +z = 3 x + 6y −z = 2 meet at angle θ , where cos θ = √ (1, 1, 1) · (1, √ 6, −1) √2 12 + 6 + (−1)2 12 + 12 + 12 √ 1+ 6−1 √√ = 38 √ 6 =√ 26 1 = 2 so θ equals π/6 radians. Parametrization of Planes So far, we have dealt with planes given as loci of linear equations. This is an implicit specification. However, there is another way to specify a plane, which is more explicit and in closer analogy to the parametrizations we have used to specify lines in space. Suppose → → − = v − +v − +v − → → v 1ı 2 3k → → − = w − +w − +w − → → w ı k 1 2 3 ...
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