Definition:Rotation (Geometry)/Space/Vector Form

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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: