# Cross product is cancellative?

1. Mar 22, 2014

### Jhenrique

If u × v = u × w, so v = w ?

2. Mar 22, 2014

### ShayanJ

No.
At first, if two vectors are equal, then they are in the same direction so lets 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.

3. Mar 22, 2014

### 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.

4. Mar 22, 2014

### mathman

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.

Last edited: Mar 22, 2014
5. Mar 22, 2014

### D H

Staff Emeritus
Not necessarily. What if u is the zero vector?

6. Mar 23, 2014

### mathman

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