\[\newcommand{\u}[1]{\boldsymbol{\mathsf{#1}}}
\renewcommand{\b}[1]{\boldsymbol{#1}}
\newcommand{\t}[1]{\textsf{#1}}
\newcommand{\m}[1]{\mathbb{#1}}
\def\RR{\bf R}
\def\bold#1{\bf #1}
\def\mbf#1{\mathbf #1}
\def\uv#1{\hat{\usf {#1}}}
\def\dl#1{\underline{\underline{#1}}}
\newcommand{\usf}[1]{\boldsymbol{\mathsf{#1}}}
\def\bs#1{\usf #1}\]
A rotation is special type of tensor (linear maps between vector spaces) takes a vector in $\mathbb{E}^n$ and gives a vector in $\mathbb{E}^n$. Specifically, the rotation tensor $\boldsymbol{R}$ satisfies the conditions
- $
\boldsymbol{R}^{\rm T}\boldsymbol{R}=\boldsymbol{I}
$
- $\text{Det}(\boldsymbol{R})=+1$