Equations defining Plane Reflection/Examples/X-Axis

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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


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


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 $x$-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:

\(\ds \map {\phi_x} P\) \(=\) \(\ds \tuple {x \map \cos {2 \times 0} + y \map \sin {2 \times 0}, x \map \sin {2 \times 0} - y \map \cos {2 \times 0} }\)
\(\ds \) \(=\) \(\ds \tuple {x \cos 0 + y \sin 0, x \sin 0 - y \cos 0}\)
\(\ds \) \(=\) \(\ds \tuple {x, -y}\) Cosine of Zero is One, Sine of Zero is Zero