I'm asked to prove $R(\vec{u} \times \vec{v}) = (R \vec{u}) \times (R \vec{v})$ for orthogonal matrices (rotations) $R$ in 3-dimensional euclidean space and use the property $\varepsilon_{ijk} \det M = \varepsilon_{i'j'k'} M_{ii} M_{jj'} M_{kk'}$.
I started with:
$$\left[ (R \vec{u}) \times (R\vec{v}) \right]_{i }= \varepsilon_{ijk} R_{j l} u_{l} R_{km}v_{m},$$
where I used $u'_{i} =(R\vec{u})_{i} = R_{ij} u_{j}$ and $(\vec{u} \times \vec{v})_{i} = \varepsilon_{ijk} u_{j} v_{k}$. So
$$\left[ (R \vec{u}) \times (R\vec{v}) \right]_{i } = \varepsilon_{ijk} R_{j l} R_{km} u_{l} v_{m} = \delta_{ir} \varepsilon_{rjk}R_{j l} R_{km} u_{l} v_{m}, $$
since $\delta_{ij} A_{jm} = A_{im}$. Now because $R$ is orthogonal, we have $R_{ik} R_{jk} = R_{ki} R_{kj} = \delta_{ij}$ from $RR^T = R^T R = I$. Then, using $\delta_{ir} = R_{is} R_{rs}$ we have
$$\left[ (R \vec{u}) \times (R\vec{v}) \right]_{i } = \delta_{ir} \varepsilon_{rjk}R_{j l} R_{km} u_{l} v_{m} = \varepsilon_{rjk} R_{is} R_{rs} R_{jl} R_{km} u_{l} v_{m} = \varepsilon_{rjk} R_{rs} R_{jl} R_{km} R_{is} u_{l} v_{m}$$
So I have something pretty close to the identity I need but not quite. If I set $\varepsilon_{rjk} R_{rs} R_{jl} R_{km} = \varepsilon_{slm} \det R$ I can finish the proof. So I have two questions:
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How can I put my proof in a form that I use the identity I need, that is, how can I make the following transformation $\varepsilon_{rjk} R_{rs} R_{jl} R_{km} \to \varepsilon_{rjk} R_{sr} R_{lj} R_{mk}$?
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Is $\varepsilon_{rjk} R_{rs} R_{jl} R_{km} R_{is} = \varepsilon_{slm} \det R$ ? If yes, how can I prove it?
Best Answer
Actually I just had to notice that $\varepsilon_{rjk} R_{rs} R_{jl} R_{km} R_{is} u_{l} v_{m} = \varepsilon_{rjk} (R^{T})_{sr} (R^{T})_{lj} (R^{T})_{mk} R_{is} u_{l} v_{m} = \varepsilon_{slm} \det (R^{T})$. That way we have
$$\varepsilon_{rjk} R_{rs} R_{jl} R_{km} R_{is} u_{l} v_{m} = \varepsilon_{rjk} (R^{T})_{sr} (R^{T})_{lj} (R^{T})_{mk} R_{is} u_{l} v_{m} = \\ = \varepsilon_{slm} \det (R^{T}) R_{is} u_{l} v_{m} =\varepsilon_{slm} \det(R) R_{is} u_{l} v_{m} = \varepsilon_{slm} R_{is} u_{l} v_{m} = \\ = R_{is} \varepsilon_{slm} u_{l} v_{m} = R_{is} \left(\vec{u} \times \vec{v}\right)_{s} = \left[ R(\vec{u} \times \vec{v})\right]_{i}. $$