Vtyjkyuk

I want to prove that $vecacdot(vecbtimesvecc) = (vecc times veca)cdotvecb$ 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

$$vecacdot(vecbtimes vecc) = a_i(vecbtimesvecc)_i = a_iepsilon_ijkb_jc_ke_i = -epsilon_jika_ib_jc_ke_i = -(vecatimesvecc)cdotvecb = (vecctimesveca)cdotvecb$$

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.

For the Levi-Civita symbols we have that, for two vectors $veca$ and $vecb$,

$vecacdotvecb = sum_ia_ib_i$

and using Einstein notation convention we can just wright $vecacdotvecb = a_ib^i$, or, we can just use that $(vecacdotvecb )_i = a_ib_i$. We also have that

$(vecatimesvecb)_i = sum_j sum_k epsilon_ijka_jb_k$

Or just $(vecatimesvecb)_i = epsilon_ijka_jb_k$. So using that you can prove the relation as you did:

$$(vecacdot(vecbtimesvecc)) = sum_ia_i(vecbtimesvecc)_i = sum_isum_jsum_ka_iepsilon_ijkb_jc_k = sum_isum_jsum_kepsilon_jkic_ka_ib_j = sum_j(vecctimesveca)_jb_j = (vecbcdot(vecctimesveca))$$

Then this is what I think was your doubt.

That's almost right, but there are some inconsistencies in your notation.

1. In the first step $acdot(btimes c)$, you have a scalar. Nothing wrong here. But note that since you begin with a scalar, you should have scalars in all the next steps.


2. In the second step $a_icdot(btimes c)_i$ you use a dot ($cdot$) between components. That is illegal. Components are numbers, and you can only use dot product between vectors. Hence, the second step should read just $a_i(btimes c)_i$


3. Since $(btimes c)_i = epsilon_ijkb_jc_k$, in step three you should have just $a_iepsilon_ijkb_jc_k$, without the vectors $e_i$. This resonates with the note in (1), where I remarked you should have just scalars and not vector expressions. Also note that this goes against the summation convention, where it is only valid to sum over pairs of indices.


4. The switch of indices and switch of sign is correct. Since, as mentioned in (3), you shouldn't write the vectors $e_i$, your concern about the index $i$ is just out of the question.

Steps 5 and 6 are indeed correct.

So, the correct derivation (with a pair of extra steps) is

$$acdot(btimes c) = a_i(btimes c)_i = a_iepsilon_ijkb_jc_k = -a_iepsilon_jikb_jc_k \= -epsilon_jika_ic_kb_j = -(atimes c)_jb_j = -(atimes c)cdot b = (ctimes a)cdot b$$