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Lecture

# lect136_5_w14.pdf

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School
University of Waterloo
Department
Mathematics
Course
MATH 136
Professor
Robert Andre
Semester
Spring

Description
Wednesday, January 15 − Lecture 5 : Cross-products in ℝ 3 Concepts: 1. cross product of two vectors in ℝ 3. 2. cross product properties In this lecture we introduce another operation on pairs of vectors in ℝ called cross product. 5.1 Definition – Given two vectors v = (a, b, c) and w = (x, y, z) in ℝ the cross product u × v is defined as follows: 5.1.1 Example – Given a = (1, 3, –2) and b = (1, 1, 5) and c = (2, –2, 3) show that the cross product (b – a) × (c – a) is (25, 7, 2). 5.1.2 Example – Given two vectors v = (a, b, c) and w = (x, y, z) verify that the cross product u × v belongs to the family of all vectors which are orthogonal to both v and w. 5.2 Theorem – Properties of cross products. Let a, b and c be vectors in ℝ . Let α ∈ ℝ. 3 CP1 a × b is a vector in ℝ . CP2 a × b is orthogonal to both a and b. (That is, a ⋅ (a × b) = 0 and b ⋅ (a × b) = 0.) CP3 a × 0 = 0 = 0 × a. CP4 a × a = 0. CP5 a × b = – (b × a) (Anti-commutative property) CP6 (αa × b) = α(a × b) = (a × αb) CP7 a × (b + c) = (a × b) + (a × c)
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