Is the Cross Product Cancellative?

Jhenrique
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If u × v = u × w, so v = w ?
 
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No.
At first, if two vectors are equal, then they are in the same direction so let's take their magnitudes. We have uv \sin\alpha=uw\sin\beta, so you have v\sin\alpha=w\sin\beta. Also \vec{v} and \vec{w} may differ in direction too.
 
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You should have been able to find a simple counterexample. E.g. if u = (1, 0, 0):

(1, 0, 0) x (x, y, z) = (0, -z, y)

So, the result only depends on y and z, and x can be anything.
 
Jhenrique said:
If u × v = u × w, so v = w ?

u × v = u × w is equivalent to u × (v-w) =0.
Therefore u is parallel to v-w, so that v-w is a multiple of u.
 
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mathman said:
u × v = u × w is equivalent to u × (v-w) =0.
Therefore u is parallel to v-w, so that v-w is a multiple of u.
Not necessarily. What if u is the zero vector?
 
D H said:
Not necessarily. What if u is the zero vector?

Quibble. The original question is pointless for u=0.
 

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