Engineering Calculus Notes 110

Engineering Calculus Notes 110 - Remark 1.7.1 If −→ v...

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98 CHAPTER 1. COORDINATES AND VECTORS (f) Show from this that AOF and CLB are similar. (g) This leads to the proportions BC BH = BC AF = BL OF = BL OD = BJ JD . Add one to both outside fractions to show that CH BH = BD JD . (h) Use this to show that ( CD ) 2 CH · HB = BD · CD JD · CD = BD · CD ( OD ) 2 . Conclude that ( CH ) 2 ( OD ) 2 = CH · HB · BD · DC. (i) Explain how this proves Heron’s formula. 1.7 Applications of Cross Products In this section we explore some useful applications of cross products. Equation of a Plane The fact that −→ v × −→ w is perpendicular to both −→ v and −→ w can be used to Fnd a “linear” equation for a plane, given three noncollinear points on it.
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Unformatted text preview: Remark 1.7.1. If −→ v and −→ w are linearly independent vectors in R 3 , then any plane containing a line ℓ v parallel to −→ v and a line ℓ w parallel to −→ w has −→ n = −→ v × −→ w as a normal vector. In particular, given a nondegenerate triangle △ ABC in R 3 , an equation for the plane P containing this triangle is −→ n · ( −→ p − −→ p ) = 0 (1.28) where −→ p = x −→ ı + y −→ + z −→ k −→ p = −−→ O A −→ n = −−→ AB × −→ AC....
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