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Crossprod of the sides of a Triangle



Show that if u,v, and w are vectors in 3D with,
u + v + w = 0
u X v = v X w = w X u


The vectors u,v and w form the directed sides of a triangle. Thus, the equality of the cross products of two of its sides (disregarding orientation) follows from the geometric definition of the cross product. The RHS rule shows that this are the products with vectors comming out of the plane.

Solution with Algebra:

Take crossprod with u on both sides of the first equation to obtain
u X v = w X u
then repeat with v istead of u to obtain the other equality.

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Carlos Rodriguez <>
Last modified: Mon Sep 18 13:48:47 EDT 2000