Equations defining Plane Reflection/Examples/X-Axis
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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 $28.4$