# How to prove that, if $A$ and $B$ are vectors, then their cross product is still a vector? [closed]

in my course of special relativity we are introducing tensors: however, before doing that, my professor sort of re-defined vectors saying that in a 3D euclidean space, $$A$$ can be called a vector if, under a rotation, it transforms like the coordinates. So, if $$x_i'=R_{ij}x_j$$ where $$R$$ is the rotation matrix and it satisfies $$R^TR=RR^T=1$$ (I'm calling $$1$$ the identity matrix), then $$A$$ is a vector if $$A_i'=R_{ij}A_j$$ Now, he told us to try to prove that, if $$A$$ and $$B$$ are vectors, then the cross product $$V=A\times B$$ is a vector too.

I know I should get the following: $$V_i'=R_{ij}V_j=R_{ij}(\varepsilon_{jlm}A_lB_m)$$ My professor also told us that the following identity is true: $$\varepsilon_{abc}R_{aj}R_{bk}R_{cm}=det(R)\varepsilon_{jkm}$$ This is what I've done: $$V_i'=(\varepsilon_{ijk}A_jB_k)'=\varepsilon_{ijk}A_j'B_k'=\varepsilon_{ijk}(R_{jl}A_l)(R_{km}B_m)$$ I have no idea how I should apply the identity my professor gave us and I don't know where that "third matrix" comes from.

• Your question is related, although not coinciding, to this physics.stackexchange.com/questions/677386/… Commented Oct 28, 2023 at 15:10
• The subsections here note some subtleties.
– J.G.
Commented Oct 28, 2023 at 18:25
• @J.G. thank you! From your link I was able to find an answer to my question
– Fede
Commented Oct 28, 2023 at 19:06