rotational invariance of cross product

Theorem
Let 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 $\mathbb{R}^{3}$ ,

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

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 u and v in that basis. Also, in the chosen basis, we denote the entries of R by $R_{ij}$ . Since 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, $j$ should be summed over $1,2,3$ . We shall use the Levi-Civita permutation symbol $\varepsilon$ to write the cross product. Then the $i$ :th coordinate of $\textbf{u}\times\textbf{v}$ equals $(\textbf{u}\times\textbf{v})^{i}=\varepsilon^{ijk}u^{j}v^{k}$ . For the $k$ 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. $\Box$

Title	rotational invariance of cross product
Canonical name	RotationalInvarianceOfCrossProduct
Date of creation	2013-03-22 13:33:53
Last modified on	2013-03-22 13:33:53
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	6
Author	matte (1858)
Entry type	Theorem
Classification	msc 15A72
Classification	msc 15A90