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.
