Can you please explain the question to me?
Question:
Let $M: \mathbb{R}^3 \to \mathbb{R}^3$, be defined to so that $M(\vec{x})$ is the mirror reflection of the vector $x$ across the plane $x_1 + 2x_2 + 3x_3 = 0$
Compute the matrix representation of $M$.
Attempt:
the normal vector to the plane is $(1,2,3)$, thus $M$ maps vector $n$ and the vector $(-n) = (-1,2,3)$.
I am stuck here. I don't know what to do next.
Best Answer
Short story: the formula for the reflection of $x$ across a plane with unit normal vector $v$ is $x-2(x^\top v) v$, or in matrix form, $Mx$ where $M=I-2vv^\top$. Here, $v=(1,2,3)/\sqrt{14}$ is the normalization of $(1,2,3)$.
Long story: geometrically, why does the formula make sense? If you decompose $x$ into $x=Proj(x)+Orth(x)$ where $Proj(x)$ is the projection onto the plane, and $Orth(x)$ is the remaining orthogonal component $x-Proj(x)$, then the reflection is clearly $Proj(x)-Orth(x) = x - 2 Orth(x)$, since you want to start from the projection $Proj(x)$ and go in the opposite direction, away from from $x$. Then, note that $Orth(x)$ can be viewed as the projection onto the normal vector of the plane, so if $v$ is this unit normal vector, then $Orth(x)=(x^\top v)v$.