Equations defining Plane Reflection/Examples/Y-Axis

Theorem
Let $\phi_y$ denote the reflection in the plane whose axis is the $y$-axis.

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

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

Proof
From Equations defining Plane Reflection:
 * $\map {\phi_\alpha} P = \tuple {x \cos 2 \alpha + y \sin 2 \alpha, x \sin 2 \alpha - y \cos 2 \alpha}$

where $\alpha$ denotes the angle between the axis and the $x$-axis.

By definition, the $y$-axis, is perpendicular to the $x$-axis

Hence $\phi_y$ can be expressed as $\phi_\alpha$ in the above equations such that $\alpha = \dfrac \pi 2$ radians.

Hence we have: