# Definition:Rotation (Geometry)/Space

## Definition

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:

Then:

- $\map {r_\theta} P = P'$

Thus $r_\theta$ is a **rotation (in space) of (angle) $\theta$ about (the axis) $O$**.

This article, or a section of it, needs explaining.In particular: 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.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

### Axis of Rotation

Let $r_\theta$ be a rotation in the Euclidean Space $\Gamma = \R^n$.

The set $A$ of points in $\Gamma$ such that:

- $\forall P \in A: \map {r_\theta} P = P$

is called the **axis of rotation** of $r_\theta$.

### Vector Form

A **space rotation** $r_\theta$ can be expressed as an axial vector $\mathbf r_\theta$ such that:

- the direction of $\mathbf r_\theta$ is defined to be its axis of rotation
- the length of $\mathbf r_\theta$ specifies its angle of rotation of $\mathbf r_\theta$ to an appropriate scale.

### Right-Hand Rule

Let $\mathbf V$ be an axial vector acting with respect to an axis of rotation $R$.

Consider a right hand with its fingers curled round $R$ so that the fingers are pointed in the direction of rotation of $\mathbf V$ around $R$.

The **right-hand rule** is the convention that the direction of $\mathbf V$ is the direction in which the thumb is pointing:

## Also see

- Results about
**geometric rotations**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**rotation**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**rotation**:**1.**