Vector Product and dot product identity: Levi-Civita symbols

I want to prove that a*(b x c) = (c x a)*b using Levi-Civita symbols, however, I am not 100% sure if my proof is correct.
Please see attached my proof

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_j*e_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.

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Well one issue I see is too many of the index i, there are three which makes the product ambiguous as which pair are summed over (since summing happens in pairs). You can fix this by omitting the unit vector as this is how the dot product works
– Triatticus
Oct 20 at 17:13

  

 

Or you can represent the dot product part as a tensor operation through use of the Kronecker Delta, that is a⋅b=aibjδija\cdot b = a_i b_j \delta_{ij}
– Triatticus
Oct 20 at 17:16

  

 

Then you can write your product as a⋅(b×c)=ai(b×c)jδij=ai(ϵjklˆejbkcl)δija \cdot (b\times c) = a_i (b \times c)_j \delta_{ij} = a_i (\epsilon_{jkl}\hat{e}_j b_k c_l) \delta_{ij}
– Triatticus
Oct 20 at 17:22

  

 

you should look at the problem of calculate the det[a.(bxc)] and compare
– janmarqz
Oct 20 at 20:31

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