Rotations/Angles naive/2,3/Section

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

Definition

A linear mapping

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

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

rotation.

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

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

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