0
$\begingroup$

Is there something analogous to the $2 \times 2$ case where we can find any reflection in the line making an angle of $\theta$ with the x axis.

1 Answers 1

0

I'll assume the plane passes through the origin, because reflection in a plane not passing through the origin is not a linear transformation and cannot be represented using a matrix alone.

Suppose the normal to the plane is the unit vector $\mathbf n$. For any point $\mathbf x$, its signed distance from the plane is $d = \mathbf n\cdot\mathbf x$. Subtracting $d\mathbf n$ from $\mathbf x$ gives you the projection of $\mathbf x$ on the plane. Subtracting it again gives you the point at the same distance from the plane but on the other side, which is the point you want. So the reflection of $\mathbf x$ is the point $\begin{align} \mathbf x' &= \mathbf x - 2(\mathbf n\cdot\mathbf x)\mathbf n \\ &= \mathbf I\mathbf x - 2\mathbf n(\mathbf n^T\mathbf x) \\ &= (\mathbf I - 2\mathbf n\mathbf n^T)\mathbf x \end{align}$ and the quantity in parentheses in the last line is the reflection matrix. By the way, this works in arbitrary dimensions, not just three.