All Problems Photometry and Geometrical Optics
Problem 5.14
Demonstrate that a light beam reflected from three mutually perpendicular plane mirrors in succession reverses its direc tion.
5.14. Suppose \(\mathbf{n}_{1}, \mathbf{n}_{2}, \mathbf{n}_{3}\) are the unit vectors of the normals to the planes of the given mirrors, and \(\mathbf{e}_{0}, \mathbf{e}_{\mathbf{1}}, \mathbf{e}_{2}, \mathbf{e}_{3}\) are the unit vectors of the incident ray and the rays reflected from the first, second, and the third mirror. Then (see the answer to the foregoing problem): \(\mathbf{e}_{1}=\mathbf{e}_{0}-2\left(\mathbf{e}_{0} \mathbf{n}_{1}\right) \mathbf{n}_{1}, \mathbf{e}_{2}=\mathbf{e}_{1}-2\left(\mathbf{e}_{1} \mathbf{n}_{2}\right) \mathbf{n}_{2}, \mathbf{e}_{3}=\mathbf{e}_{2}-2\left(\mathbf{e}_{2} \mathbf{n}_{3}\right) \mathbf{n}_{3}\) Summing termwise the left-hand and right-hand sides of these expressions, it can be readily shown that \(\mathbf{e}_{3}=-\mathbf{e}_{\boldsymbol{r}}\).