# simplify ax[ax(axb)]dotc

• Feb 11th 2013, 09:56 PM
lytwynk
simplify ax[ax(axb)]dotc
Hello, we were given a few practice problems in class the other day and one of them asks...

Simplify a x [a x (a x b)] dot c (where the x refers to cross product)

I have tried to simplify it but have ended up with about a page and a half of computation that I don't know what to do with.

I imagine there is a trick to this but I am clueless. If anyone can help I would appreciate it.
• Feb 11th 2013, 10:02 PM
chiro
Re: simplify ax[ax(axb)]dotc
Hey lytwynk.

Hint: Use the fact that the cross product is associative and re-group [a x (a x b)] = (a x a) x b.
• Feb 11th 2013, 10:11 PM
lytwynk
Re: simplify ax[ax(axb)]dotc
But I thought the cross product was not associative for example a x (b x c) does not equal (a x b) x c... At least that's what my text says.
• Feb 11th 2013, 10:24 PM
chiro
Re: simplify ax[ax(axb)]dotc
Sorry you are right: it is not associative.

Try using the Jacobi identity and you should get some cancellations:

Jacobi identity - Wikipedia, the free encyclopedia
• Feb 12th 2013, 07:00 AM
hollywood
Re: simplify ax[ax(axb)]dotc
You could also use the identity $A \times (B \times C) = (A \cdot C)B - (A \cdot B)C$.

- Hollywood