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Three mutually perpendicular mirrors reverse a ray's direction when it encounters all three. With orthonormal normals $a,b,c$, write the reflections as $I-2P_a$, $I-2P_b$, $I-2P_c$. Their orthogonal projection matrices have pairwise products zero and sum to $I$ by the resolution of the identity in an orthonormal basis. The product of the three reflections is therefore $-I$. It reverses every incident direction independently of the order in which the mirrors are met. Real apertures restrict which incoming rays actually reach all three reflecting faces. Such retroreflectors provide the returning-light geometry in lunar laser ranging.

For an arbitrary vector $\mathbf{x}$, expansion in the orthonormal basis gives $x_i=(a_i{a}_j+b_i{b}_j+c_i{c}_j)x_j$, with summation over $j$. Since this holds for every $\mathbf{x}$, the resolution of the identity in an orthonormal basis isLet $P_a=\mathbf{a}{\mathbf{a}}^T$, and similarly for $b,c$. The three orthogonal projection matrices sum to $I$ and have pairwise products zero. ThusAny ray meeting all three ideal perpendicular mirrors therefore returns in the opposite direction, independently of reflection order. This is the geometry of a corner reflector, a kind of retroreflector used in lunar laser ranging.