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Lecture

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University of Waterloo

Mathematics

MATH 136

Robert Andre

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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