Triple product proof

Question

I am trying to prove the identity $a\times(b\times c)=\langle a,c\rangle b-\langle a,b\rangle c$

Like other proofs (such as this one) , I have managed to find that $a\times(b\times c)=\lambda\langle a,c\rangle\cdot b-\lambda\langle a,b\rangle\cdot c$

That is true for some $\lambda \in \mathbb R$, in the proof that I mentioned above, they state that this constant $\lambda$ is the same for all three vectors $a,b,c\in\mathbb R^3$.

I just cant understand why this constant is the same for every trio, I would appreciate any explanation, Thanks.

The easiest is to use the identity $(a \times b)^i = \epsilon^{ijk} a^j b^k$ where $\epsilon^{ijk}$ is the Levi-Civita symbol. — Mathphys meister, Jul 22 '20 at 06:07
Quora is not the place where I would search for answers for technical questions. The most straightforward way to prove the identity is to just represent all vectors in the standard Euclidean basis. — Dmitry, Jul 22 '20 at 06:15
In particular, @Mathphysmeister's approach uses $\epsilon^{ijk}\epsilon_{ilm}=\delta^j_l\delta^k_m-\delta^k_l\delta^j_m$. This has a proof in terms of antisymmetries using logic similar to one part of my answer. — J.G., Jul 22 '20 at 07:53
@J.G. this is indeed right. Thanks for elaborating on my comment. — Mathphys meister, Jul 22 '20 at 11:40
Does this answer your question? Do the BAC-CAB identity for triple vector product have some intepretation? — Ted Shifrin, Jul 23 '20 at 03:15
I would love to help but I don't understand what you mean by "this constant is the same for all three vectors ,,∈$ℝ^3$"? — underdisplayname, Jul 27 '20 at 03:47

Triple product proof

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