- #1
Marin
- 193
- 0
Hi there!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 productprove: [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 problemsIf you have any idea, I'd be glad to see it :)With regards,
marin
Let [tex]\vec a,\ \vec b\in\mathbb{R^3}[/tex] , let M be in SO(3) and x be the cross productprove: [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 problemsIf you have any idea, I'd be glad to see it :)With regards,
marin