# Rotations/Angles naive/2,3/Section

A rotation of the real plane ${\displaystyle {}\mathbb {R} ^{2}}$ around the origin, given the angle ${\displaystyle {}\alpha }$ counterclockwise, maps ${\displaystyle {}{\begin{pmatrix}1\\0\end{pmatrix}}}$ to ${\displaystyle {}{\begin{pmatrix}\cos \alpha \\\sin \alpha \end{pmatrix}}}$ and ${\displaystyle {}{\begin{pmatrix}0\\1\end{pmatrix}}}$ to ${\displaystyle {}{\begin{pmatrix}-\sin \alpha \\\cos \alpha \end{pmatrix}}}$. Therefore, plane rotations are described in the following way.

## Definition

${\displaystyle D(\alpha )\colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ^{2},}$

which is given by a rotation matrix ${\displaystyle {}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}}$ (with some ${\displaystyle {}\alpha \in \mathbb {R} }$)with respect to the standard basis is called

rotation.

A space rotation is a linear mapping of the space ${\displaystyle {}\mathbb {R} ^{3}}$ in itself around a rotation axis (a line through the origin) with an certain angle ${\displaystyle {}\alpha }$. If the vector ${\displaystyle {}v_{1}\neq 0}$ defines the axis, and ${\displaystyle {}u_{2}}$ and ${\displaystyle {}u_{3}}$ are orthogonal to ${\displaystyle {}v_{1}}$ and to each other, and all have length ${\displaystyle {}1}$, then the rotation is described by the matrix

${\displaystyle {\begin{pmatrix}1&0&0\\0&\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\0&\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}}$

with respect to the basis ${\displaystyle {}v_{1},u_{2},u_{3}}$.