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:

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

$\map {r_\alpha} P = \tuple {x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha}$

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:

$\ds \map {r_\Box} P$

$=$

$\ds \tuple {x \cos \dfrac \pi 2 - y \sin \dfrac \pi 2, x \sin \dfrac \pi 2 + y \cos \dfrac \pi 2}$

$\ds $

$=$

$\ds \tuple {-y, x}$

$\blacksquare$