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:

\(\ds \map {\phi_x} P\)

\(=\)

\(\ds \tuple {x \map \cos {2 \times 0} + y \map \sin {2 \times 0}, x \map \sin {2 \times 0} - y \map \cos {2 \times 0} }\)

\(\ds \)

\(=\)

\(\ds \tuple {x \cos 0 + y \sin 0, x \sin 0 - y \cos 0}\)

\(\ds \)

\(=\)

\(\ds \tuple {x, -y}\)

Cosine of Zero is One, Sine of Zero is Zero

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.4$