So I was wondering how you would determine the matrix of the reflections in a plane in ℝ3. For example if you have a function like $x_1 + x_2 + x_3 = 0$ then how would you find the matrix of the reflection? I know that for lines what I did was I found a point in the line (which I call n) and then I used that in this formula:
$e_1 -2\dfrac{e_1 · n}{||n||^2}n$ for e1 and e2 where e1 and e2 = $\begin{bmatrix}1\\0\end{bmatrix}$ and $\begin{bmatrix}0\\1\end{bmatrix}$ respectively. It worked for lines no matter what n vector I picked but for planes I'm using the same formula (except using e1, e2, and e3) but the answer seems to change depending on what n value I picked and none of the answers seem to be the correct one. So do I have to do something differently for finding reflections in planes as opposed to lines?
Also the answer to
$x_1 + x_2 + x_3 = 0$ is $1/3\begin{bmatrix}1&-2&-2\\-2&1&-2\\-2&-2&1\end{bmatrix}$ but I can't seem to get that answer using the above formula.
Best Answer
If $n = (a, b, c)$ is a unit vector orthogonal to the plane $P$, then reflection in $P$ sends $(x, y, z)$ to $$ (x, y, z) - 2\operatorname{proj}_{n}(x, y, z) = (x, y, z) - 2(ax + by + cz)(a, b, c). $$ The standard matrix is $$ \left[\begin{array}{@{}ccc@{}} 1 - 2a^{2} & -2ab & -2ac \\ -2ab & 1 - 2b^{2} & -2bc \\ -2ac & -2bc & 1 - 2c^{2} \\ \end{array}\right]. $$