I want to prove that $\vec{a}\cdot(\vec{b}\times\vec{c}) = (\vec{c} \times \vec{a})\cdot\vec{b}$ using the Levi-Civita symbols, however, I am not $100$% sure if my proof is correct.
Please see attached my proof, see the image 
Or see the (using MathJax) equations below
$$\vec{a}\cdot(\vec{b}\times \vec{c}) = a_i(\vec{b}\times\vec{c})_i = a_i\epsilon_{ijk}b_jc_ke_i = -\epsilon_{jik}a_ib_jc_ke_i = -(\vec{a}\times\vec{c})\cdot\vec{b} = (\vec{c}\times\vec{a})\cdot\vec{b}$$
My main concern is that when I change the indices for epsilon from $(i,j,k)$ to $(j,i,k)$, should I also change the index for $e$ vector from $i$ to $j$ as well? It's just in my proof I assume that $b_je_i$ gives vector $b$ and I do not know if I can state that given the different indices.
Thank you in advance and I hope this all does not sound too confusing.
Best Answer
For the Levi-Civita symbols we have that, for two vectors $\vec{a}$ and $\vec{b}$,
and using Einstein notation convention we can just wright $\vec{a}\cdot\vec{b} = a_ib^i$, or, we can just use that $(\vec{a}\cdot\vec{b} )_i = a_ib_i$. We also have that
Or just $(\vec{a}\times\vec{b})_i = \epsilon_{ijk}a_jb_k$. So using that you can prove the relation as you did:
$$(\vec{a}\cdot(\vec{b}\times\vec{c})) = \sum_ia_i(\vec{b}\times\vec{c})_i = \sum_i\sum_j\sum_ka_i\epsilon_{ijk}b_jc_k = \sum_i\sum_j\sum_k\epsilon_{jki}c_ka_ib_j = \sum_j(\vec{c}\times\vec{a})_jb_j = (\vec{b}\cdot(\vec{c}\times\vec{a}))$$
Then this is what I think was your doubt.