Engineering Calculus Notes 110

Engineering Calculus Notes 110 - Remark 1.7.1. If v and w...

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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 v parallel to v and a line 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 + y + z k p = O A n = AB AC....
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