Equations defining Plane Reflection/Examples/X-Axis

Theorem
Let $\phi_x$ denote the reflection in the plane whose axis is the $x$-axis.

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

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

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 $x$-axis, being coincident with itself, is at a zero angle with itself.

Hence $\phi_x$ can be expressed as $\phi_\alpha$ in the above equations such that $\alpha = 0$.

Hence we have: