I would prove this using the Levi-Civita density symbol for cross products. It looks like and it's defined this way:
- from Classical Dynamics of Particles and Systems, 4th Ed., p. 25-26, by Marion and Thornton.
An even permutation has an even number of exchanges of position of two symbols. Cyclic permutations (for example, ) are always even. Thus
The cool thing about the Levi-Civita symbol is that you can write cross-products with it explicitly (or at least components of it explicitly):
So, getting to your first question, I can write the cross product as
Then I get to say that
The rest of the proof consists of massaging this last expression to equal your original RHS.
As for your part 2, do you mean that you need to show