Tensor notation proofs in Marion and Thornton section 1.6

[Winzer]

Homework Statement

I am having trouble understanding the proofs in Marion and Thorton [Newest Edition]. The section where he goes through proof of products in tensor notation. An example is page 26 example 1.6. I don't get the switching of the indices on the very last part. Also can someone prove to me that $$\vec{A}\times \vec{B}=-\vec{b}\times \vec{A}$$ using tensor summation notation. Want to make sure I am doing it right.

 

[Pacopag]

I'm not sure that this is the answer you are looking for, but I think you can just write out both sides of the equation using tensor notation (i.e. using the Levi-Civita symbol), and remember that Levi-Civita is antisymmetric.