Hi there!(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**