If $[a]_\times$ is a matrix such that $[a]_\times b$ = $a \times b$, and $R$ is a (rotation) Matrix, how to simplify $[Ra]_\times$?

Question

Suppose $a \in \mathbb{R}^3$, $b \in \mathbb{R}^3$ and $R \in \mathbb{R}^{3\times3}$.

Defining the operation $[a]_\times = \begin{bmatrix} 0 & - a_{z} & a_{y}\\a_{z} & 0 & - a_{x}\\- a_{y} & a_{x} & 0 \end{bmatrix}$ so that:

$[a]_\times b$ = $a \times b$

Now suppose $c = R a$. Is there any way to simplify the following expression:

$[c]_\times = [R a]_\times$ expressing it as some kind of product with $[a]_\times$ and $R$?

I have tried to find this relationship but I failed to find anything. So if someone can point to me where I can find a relationhip for this (in case it exists), it would mean a lot!

Answer 1

If you rotate both $a$ and $b$ by the matrix $R$ then it is evident (without proof) that

$ (R a) \times (R b) = R (a \times b ) $

The left hand side $[ R a ]_\times (R b) $ and the right hand side is $R [a]_\times b$. Thus,

$[ R a ]_\times R b = R [a]_\times b$

And since this is true for any vector $b$, it follows that

$[ R a ]_\times R = R [a]_\times $

So that,

$[ R a ]_\times = R [a]_\times R^T $