Law of reflection

Remarks

The vector form says that $d_r$ is the image of $d$ under the reflection in the tangent plane of the surface. The tangential part of $d$ is kept and the normal part changes sign.

One proof uses Fermat's principle for a plane mirror. Reflecting the end point $B$ in the mirror to a point $B'$ turns a path $A \to x \to B$ into a path $A \to x \to B'$ of the same length. This path is shortest when it is a straight line. That line makes equal angles with the normal on both sides.

Example: a ray with direction $d = (1, -1, 0)/\sqrt{2}$ hits the mirror $y = 0$ with normal $\nu = (0, 1, 0)$. Then $d \cdot \nu = -1/\sqrt{2}$ and $d_r = (1, 1, 0)/\sqrt{2}$: both rays make the angle $45^\circ$ with the normal.

Going deeper

For a student

The law of reflection, with Snell's law, is all that ray tracing needs at a surface. For curved mirrors it is applied with the normal at each point, which gives the focusing of a parabolic mirror: rays parallel to its axis reflect through its focus. In wave optics it follows from matching the phases of the incident and reflected plane waves along the surface.