\begin{definition}[Definition:Rotation (Geometry)/Space]
A '''rotation''' $r_\theta$ in space is an isometry on the Euclidean Space $\Gamma = \R^3$ as follows.
Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that:
:$\forall P \in AB: \map {r_\theta} P = P$
That is, all points on $AB$ map to themselves.
Let $P \in \Gamma$ such that $P \notin AB$.
Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$.
Let a straight line $OP'$ be constructed perpendicular to $AB$ such that:
:$(1): \quad OP' = OP$
:$(2): \quad \angle POP' = \theta$ such that $OP \to OP'$ is in the anticlockwise direction:
:400px
Then:
:$\map {r_\theta} P = P'$
Thus $r_\theta$ is a '''rotation (in space) of (angle) $\theta$ about (the axis) $O$'''.
{{explain|In this context, the "anticlockwise direction" is not well defined. This page is to be revisited with more rigour by someone who has a better grasp of exactly what the concepts are.}}
\end{definition}