Swapping indices in cross product

Question

Is doing $$\epsilon_{ijk}a_jb_k=-\epsilon_{ikj}a_jb_k$$the same as doing$$\epsilon_{ijk}a_jb_k=-\epsilon_{ijk}a_kb_j$$and is this true?: $$\epsilon_{ijk}a_jb_k=\epsilon_{ijk}b_ka_j$$ thanks

Answer 1

Short answer: Yes, and yes.

For the first one, you are allowed to rename dummy (contracted) indices, and this doesn't change anything. They are, after all, just summation indices, and summation indices can be named whatever we want. So we have $$ -\epsilon_{ikj}a_jb_k=-\epsilon_{ipj}a_jb_p\\ =-\epsilon_{ipk}a_kb_p=-\epsilon_{ijk}a_kb_j $$ where each equality is just a renaming of dummy indices. This has the net effect of swapping two dummy indices, which therefore is also allowed and doesn't change the value.

The second one is also true. For each term of the implicit sum, $a_j$ and $b_k$ represent real numbers. And real numbers commute: $a_jb_k=b_ka_j$. If this seems strange to you, then write out the entire sum instead of relying on the implicit index contraction, and see that this is true.