# Cross product and matrix multiplication

1. Jul 23, 2012

### haael

Suppose that we have a cross-product of vectors.

$$a × b = c$$

Now suppose that we have an orthogonal matrix M. Is it true that

$$(M a) × (M b) = M c$$

?

My intuition is that here we are moving to another coordinate system and performing a cross product in this new system. I can't find an answer in google, so I'm posting here.

2. Jul 23, 2012

### homeomorphic

Give it determinant 1, otherwise it will reverse orientation and screw up your right-hand rule. Then, it works. The transformation will preserve the norms of the vectors, and it will also map orthogonal vectors to orthogonal vectors. So, far that leaves only two possibilities when you take the cross product. Which is why you make it have determinant one. Then, the right hand rule is taken care of.

3. Jul 23, 2012

### algebrat

It looks like proof goes through if you check (M_1a_1+...)x(M_1b_1+...), distribute (like FOIL) and use orthonormality of columns of M. I didnt check if the determinant = 1 condition mentioned above is necessary.

Last edited: Jul 23, 2012
4. Jul 23, 2012

### micromass

Staff Emeritus
It should be necessary. If the determinant is -1, then it is false since orientation is changed.