Is the Cross Product Cancellative?

[Jhenrique]

If u × v = u × w, so v = w ?

 

[ShayanJ]

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.

 

[PeroK]

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.

 

[mathman]

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.

 

[D H]

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?

 

[mathman]

Not necessarily. What if u is the zero vector?

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