# Math Help - simplify ax[ax(axb)]dotc

1. ## 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.

2. ## 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.

3. ## 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.

4. ## 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

5. ## 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