Equations defining Plane Rotation/Examples/Right Angle

Theorem
Let $r_\Box$ be the rotation of the plane about the origin through a right angle.

Let $P = \tuple {x, y}$ be an arbitrary point in the plane

Then:
 * $\map {r_\Box} P = \tuple {y, -x}$

Proof
From Equations defining Plane Rotation:
 * $\map {r_\alpha} P = \tuple {x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha}$

where $\alpha$ denotes the angle of rotation.

Hence $r_\Box$ can be expressed as $r_\alpha$ in the above equations such that $\alpha = \dfrac \pi 2$.

Hence we have: