Rotation

Theorem

Let $r_\alpha$ be the rotation of the plane about the origin through an angle of $\alpha$.

That is, let $r_\alpha: \R^2 \to \R^2$ be the mapping defined as:

$\forall x \in \R^2: r_\alpha \left({x}\right) = \text { the point into which a rotation of } \alpha \text{ carries } x$

Then $r_\alpha$ is a linear operator determined by the ordered sequence:

$\left({\cos \alpha, -\sin \alpha, \sin \alpha, \cos \alpha}\right)$

Proof

• Let $\left({\lambda_1, \lambda_2}\right) = \left({\rho \cos \sigma, \rho \sin \sigma}\right)$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle r_\alpha \left({\lambda_1, \lambda_2}\right)\)	\(=\)	\(\displaystyle \left({\rho \cos \alpha \cos \sigma - \rho \sin \alpha \sin \sigma, \rho \sin \alpha \cos \sigma + \rho \cos \alpha \sin \sigma}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left({\lambda_1 \cos \alpha - \lambda_2 \sin \alpha, \lambda_1 \sin \alpha + \lambda_2 \cos \alpha}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

The result follows from Linear Operator on the Plane.

Sources

• Seth Warner: Modern Algebra (1965): $\S 28$: Example $28.2$