Simplification using suffix notation

Question

I have a question and I need to simplify $(a\times b)+c=(a\cdot b)b-d$ using suffix notation.

$(a\times b)_i=\epsilon_{ijk}a_jb_k$ and $(a\cdot b)_i=a_jb_j$

So far I have got to $\epsilon_{ijk}a_jb_k+c_i=a_jb_jb_i-d_i$. Is there any other simplification I can do?

Thanks in advance

Answer 1

It's a matter of taste and goal in mind, but you can also do this: $$ c_i +d_i = a_jb_jb_i - \epsilon_{ijk}a_jb_k=a_j\delta_{jk}b_kb_i-\epsilon_{ijk}a_jb_k = (\delta_{jk}b_i-\epsilon_{ijk})a_jb_k $$