Einstein summation convention proof

[tigger88]

Homework Statement

Using the Einstein summation convention, prove:

A$$\bullet$$B$$\times$$C = C$$\bullet$$A$$\times$$B

Homework Equations

The Attempt at a Solution

I tried to follow an example from my notes, but I don't entirely understand it. Would it be possible to find out if what I've done (below) is correct, or where I went wrong?

(B$$\times$$C)$$_{i}$$ = $$\epsilon$$$$_{ijk}$$B$$_{j}$$C$$_{k}$$

(A$$\bullet$$B$$\times$$C)$$_{i}$$ = $$\epsilon$$$$_{ijk}$$A$$_{i}$$B$$_{j}$$C$$_{k}$$

= $$\epsilon$$$$_{kij}$$A$$_{i}$$B$$_{j}$$C$$_{k}$$

=C$$\bullet$$A$$\times$$B

Thanks!
PS. I'm not very familiar with Latex and couldn't get the symbols to line up properly.. sorry! They should all be subscripts.

 

[Dick]

I think that's about right. The thing is just to show that e_{ijk}=e_{jki}, right?