Equations defining Plane Reflection/Examples/Y-Axis

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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


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


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:

\(\ds \map {\phi_x} P\) \(=\) \(\ds \tuple {x \map \cos {2 \dfrac \pi 2} + y \map \sin {2 \dfrac \pi 2}, x \map \sin {2 \dfrac \pi 2} - y \map \cos {2 \dfrac \pi 2} }\)
\(\ds \) \(=\) \(\ds \tuple {x \cos \pi + y \sin \pi, x \sin \pi - y \cos \pi}\)
\(\ds \) \(=\) \(\ds \tuple {-x, y}\) Cosine of Straight Angle, Sine of Straight Angle