HW 12.4-solutions

# 1 true 2 false correct 1 d a ab 2 d a b correct

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Unformatted text preview: Let ℓ be the line passing through points Q, R and P a point not on ℓ as shown in morrell (mm59638) – HW 12.4 – radin – (56025) ℓ R 5 keywords: distance, distance from line, cross product, vectors D 011 a d Q 10.0 points If u, v and w are non-zero vectors such that θ (u × v ) × w = 0 , P b then u and v must be parallel. Express the distance d from P to ℓ in terms of the vectors − − → a = QR , − − → b = QP . True or False? 1. TRUE 2. FALSE correct 1. d = | a| a·b 2. d = |a × b| correct | a| Explanation: If w = 0, then |b| 3. d = |a × b| 4. d = a·b | a| (u × v ) × w = 0 if and only if u × v = 0 (in which case u and v are parallel) or w and u × v are parallel. But if w is orthogonal to both u and v, then w is parallel to u × v. For example, when u = i, v = j, w = k, 5. d = |a × b| |b| then u is perpendicular, not parallel, to v, while 6. d = | a| |a × b| ( u × v ) × w = ( i × j) × k = k × k = 0 . Explanation: Graphically, d is the length of the perpen−→ − dicular P D from P to ℓ shown in the ﬁgure. Now by right angle trigonometry, Consequently, the statement is FALSE . d = |b| sin θ . 012 10.0 points On the other hand, |a × b| = |a| |b| sin θ ; i.e., But then |a × b| . |b| sin θ = | a| d= |a × b| . | a| Which of the following expressions are welldeﬁned for all vectors a, b, c, and d? I II a · ( b × c) , (a · b) × (c · d) , III a × (b × c) . 1. II only morrell (mm59638) – HW 12.4 – radin – (56025) 6 2. III only 3. A and C only 3....
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## This document was uploaded on 04/04/2014.

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