$\newcommand{\Reals}{\mathbb{R}}$I have an equation of a line: $4x - 3y = 0$. Let $S : \Reals^2 \to \Reals^2$ be reflection through that line, and let $P : \Reals^2 \to \Reals^2$ be projection onto that line.
I want to find a vector that represents the reflection through that line and a vector that represents the projection onto that line. Are there any formulas for these?
Let $\vec{u}=\left[\begin{smallmatrix}\tfrac{3}{5} \\ \tfrac{4}{5}\end{smallmatrix}\right]$ be the unit vector on that line and let $\vec{v}=\left[\begin{smallmatrix}\tfrac{4}{5} \\ -\tfrac{3}{5}\end{smallmatrix}\right]$ be a unit vector orthogonal to $\vec{u}$. Any vector $\vec{a}\in {\mathbb R}^{2}$ and be uniquely repersented as $\vec{a}=\lambda \vec{u}+\mu \vec{v}$ with $\lambda, \mu \in {\mathbb R}$. Then $$ S\vec{a}=\lambda \vec{u}-\mu \vec{v}\quad \text{and}\quad P\vec{a}=\lambda \vec{u}. $$