PlanetMath: rotational invariance of cross product

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rotational invariance of cross product

(Theorem)

Theorem
Let $\textbf{R}$ be a rotational $3\times 3$ matrix, i.e., a real matrix with $\det \textbf{R}= 1$ and $\textbf{R}^{-1} = \textbf{R}^T$ . Then for all vectors $\textbf{u},\textbf{v}$ in $\mathbb{R}^3$ ,

$\displaystyle \textbf{R}\cdot (\textbf{u}\times \textbf{v}) = (\textbf{R}\cdot \textbf{u})\times (\textbf{R}\cdot \textbf{v}).$

Proof. Let us first fix some right hand oriented orthonormal basis in $\mathbb{R}^3$ . Further, let $\{u^1,u^2,u^3\}$ and $\{v^1,v^2,v^3\}$ be the components of $\textbf{u}$ and $\textbf{v}$ in that basis. Also, in the chosen basis, we denote the entries of $\textbf{R}$ by $R_{ij}$ . Since $\textbf{R}$ is rotational, we have $R_{ij} R_{kj} = \delta_{ik}$ where $\delta_{ik}$ is the Kronecker delta symbol. Here we use the Einstein summation convention. Thus, in the previous expression, on the left hand side, should be summed over . We shall use the Levi-Civita permutation symbol $\varepsilon$ to write the cross product. Then the :th coordinate of $\textbf{u}\times \textbf{v}$ equals $(\textbf{u}\times \textbf{v})^i = \varepsilon^{ijk} u^j v^k$ . For the th component of $(\textbf{R}\cdot \textbf{u})\times (\textbf{R}\cdot \textbf{v})$ we then have

$\displaystyle ((\textbf{R}\cdot \textbf{u})\times (\textbf{R}\cdot \textbf{v}))^k$	$\displaystyle =$	$\displaystyle \varepsilon^{imk} R_{ij} R_{mn} u^j v^n$
	$\displaystyle =$	$\displaystyle \varepsilon^{iml} \delta_{kl} R_{ij} R_{mn} u^j v^n$
	$\displaystyle =$	$\displaystyle \varepsilon^{iml} R_{kr} R_{lr} R_{ij} R_{mn} u^j v^n$
	$\displaystyle =$	$\displaystyle \varepsilon^{jnr} \det \textbf{R}\, R_{kr} u^j v^n.$

The last line follows since $\varepsilon^{ijk} R_{im} R_{jn} R_{kr} = \varepsilon^{mnr}\varepsilon^{ijk} R_{i1} R_{j2} R_{k3} = \varepsilon^{mnr} \det \textbf{R}$ . Since $\det \textbf{R}= 1$ , it follows that

$\displaystyle ((\textbf{R}\cdot \textbf{u})\times (\textbf{R}\cdot \textbf{v}))^k$	$\displaystyle =$	$\displaystyle R_{kr} \varepsilon^{jnr} u^j v^n$
	$\displaystyle =$	$\displaystyle R_{kr} (\textbf{u}\times \textbf{v})^r$
	$\displaystyle =$	$\displaystyle (\textbf{R}\cdot \textbf{u}\times \textbf{v})^k$

as claimed. $\Box$

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Cross-references: line, coordinate, cross product, Levi-Civita permutation symbol, left hand side, expression, Einstein summation convention, Kronecker delta, basis, components, orthonormal basis, oriented, right, fix, vectors, real, matrix
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This is version 3 of rotational invariance of cross product, born on 2003-04-09, modified 2003-06-10.
Object id is 4171, canonical name is RotationalInvarianceOfCrossProduct.
Accessed 3245 times total.

Classification:

AMS MSC:	15A72 (Linear and multilinear algebra; matrix theory :: Vector and tensor algebra, theory of invariants)
	15A90 (Linear and multilinear algebra; matrix theory :: Applications of matrix theory to physics)

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