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I'm trying to prove the following obvious statement, but am somehow stuck :(

Let [tex]\vec a,\ \vec b\in\mathbb{R^3}[/tex] , let M be in SO(3) and x be the cross product

prove: [tex]M(\vec a\times\vec b)=M\vec a\times M\vec b[/tex]

I tried using the epsilon tensor, as in physics, but it doesn't really produce an opportunity, as you can convince yourself:

[tex](M\vec a\times M\vec b)_i=\varepsilon_{ijk}(M\vec a)_j(M\vec b)_k=\varepsilon_{ijk}M_{jl} a_l M_{km}b_m...[/tex]

where as usual summation is over repeated indices

Now I want to use the fact that M is orthogonal, i.e.

[tex]M_{ij}M_{jk}^t=\delta_{ik}[/tex],

and preserves the orientation of the basis but I don't know where exactly this has to come into the proof...

What I want to end up with is

[tex]...=M_{ij}\varepsilon_{jkl}a_kb_l=(M(\vec a\times\vec b))_i[/tex]

The statement seems to me obvious and can be envisioned very quickly by the right-hand-rule; I don't know why establishing it makes real problems

If you have any idea, I'd be glad to see it :)

With regards,

marin

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# Cross product invariant under SO(3)-matrices?

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