Equations defining Plane Reflection/Examples/Y-Axis
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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:
\(\ds \map {\phi_x} P\) | \(=\) | \(\ds \tuple {x \map \cos {2 \dfrac \pi 2} + y \map \sin {2 \dfrac \pi 2}, x \map \sin {2 \dfrac \pi 2} - y \map \cos {2 \dfrac \pi 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x \cos \pi + y \sin \pi, x \sin \pi - y \cos \pi}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {-x, y}\) | Cosine of Straight Angle, Sine of Straight Angle |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations: Example $28.4$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): reflection: 2. (in a line)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): reflection: 2. (in a line)