Chapter 1: Vectors, Lines and Planes
Section 1.5: Applications of Vector Products
If A≠0, show that A×B=A×C and A·B=A·C together imply that B=C.
Equate the lengths of the cross products, so that A B sinθ1 = A C sinθ2, where θ1 is the angle between A and B, and θ2 is the angle between A and C. This gives
B∥C∥ = sinθ2sinθ1
The equality A·B=A·C gives ABcos(θ1) = ACcos(θ2), from which it follows that
B∥C∥ = cosθ2cosθ1
Therefore sinθ2sinθ1 = cosθ2cosθ1 so sinθ2cosθ1−cosθ2sinθ1=0=sinθ2−θ1, from which it follows that θ2=θ1. Consequently, B = C, and since the angles these vectors make with the fixed vector A are the same, the vectors B and C must be the same.
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