English: Animation showing the reflection of the standard basis vectors ${\displaystyle {\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3}$ and the unit cube across the plane ${\displaystyle \mathrm{col}(U)\subset {ℝ}^3}$.

The orthogonal projection onto the plane is

$${\displaystyle P=U(U^{T}U{)}^{-1}{U}^T,}$$

and the corresponding reflection matrix is

$${\displaystyle R=2P-I.}$$

The animation shows the original vectors and cube, their orthogonal projections onto the plane, and their reflected images. Vectors lying in the plane remain fixed, while components perpendicular to the plane are reversed.

The animation is parametrized by gradually reducing the component perpendicular to the plane. At the midpoint, this component becomes zero, so the vectors and cube reach their orthogonal projections onto the plane; continuing the same motion reverses the perpendicular component and produces the reflected configuration.