Exercise:
A light ray is reflected in turn by three plane mirrors mutually at right angles to each other. Show that the incident and the reflected rays are parallel.

Solution:
center tikzpicture Latex line width.pt x.cm.cm y.cm-.cm zcmcm % --- The three mirror planes corner of a cube meeting at origin O --- coordinate O at ; % Mirror : the xy-plane floor filldrawfillblue! drawblue! fill opacity. -- -- -- -- cycle; % Mirror : the yz-plane left wall filldrawfillgreen! drawgreen! fill opacity. -- -- -- -- cycle; % Mirror : the xz-plane back wall filldrawfillorange! draworange! fill opacity. -- -- -- -- cycle; % Axis labels for clarity nodeblue! at .. Mirror xy-plane; nodegreen!!black at .. Mirror yz-plane; nodeorange!!black at .. Mirror xz-plane; % --- The bouncing ray: incoming - hits mirror P - mirror P - mirror P - outgoing --- coordinate Pin at ...; coordinate P at ..; % reflects off floor coordinate P at ..; % reflects off left wall coordinate P at ..; % reflects off back wall coordinate Pout at .-..; draw- thick red Pin -- P nodemidway above right black incoming; draw- thick red P -- P; draw- thick red P -- P; draw- thick red P -- Pout nodemidway above right black outgoing; fill P circle .pt nodebelow P_; fill P circle .pt nodeleft P_; fill P circle .pt nodeabove right P_; fill O circle .pt nodebelow left O; % small dashed arrow to asize the two rays are parallel drawdashed gray Pin -- ++...; drawdashed gray Pout -- ++-.-.-.; tikzpicture center noindent textbfSetup. Since the three mirrors are mutually perpicular we may choose a Cartesian coordinate system whose axes are ed with their normals. Concretely place the three mirrors along the coordinate planes: textMirror : the xytext-plane qquad textMirror : the yztext-plane qquad textMirror : the xztext-plane. Let the incident ray travel in the direction of the unit vector mathbfd d_x d_y d_z. medskip noindent textbfKey fact: reflection off a coordinate plane. When a ray reflects off a plane the component of its direction vector normal to that plane is reversed while the components parallel to the plane are unchanged. Concretely: itemize item Reflection off the xy-plane normal along z ss d_xd_yd_z mapsto d_xd_y-d_z. item Reflection off the yz-plane normal along x ss d_xd_yd_z mapsto -d_xd_yd_z. item Reflection off the xz-plane normal along y ss d_xd_yd_z mapsto d_x-d_yd_z. itemize medskip noindent textbfApplying the three reflections in turn. Suppose the ray strikes Mirror first then Mirror then Mirror the order does not matter as shown below. Starting from mathbfd d_x d_y d_z: d_x d_y d_z ;xrightarrowtextMirror xytext; d_x d_y -d_z ;xrightarrowtextMirror yztext; -d_x d_y -d_z ;xrightarrowtextMirror xztext; -d_x -d_y -d_z. Each mirror flips exactly one component's sign and since there are three mirrors covering the three indepent directions xyz all three components get flipped exactly once. Hence the final direction vector is mathbfd' -d_x -d_y -d_z -mathbfd. medskip noindent textbfOrder indepence. Since each reflection only negates the sign of the component along its own normal direction and multiplication of signs is commutative the three sign flips can be applied in any order and still yield mathbfd' -mathbfd. Thus the result does not dep on which mirror the ray hits first. medskip noindent textbfConclusion. The outgoing direction mathbfd' -mathbfd is exactly the negative of the incoming direction mathbfd. Two vectors that are scalar multiples of one another here by - are by definition parallel. Therefore the ray leaving the corner reflector after three reflections travels exactly antiparallel to -- and hence parallel to -- the original incident ray regardless of the incidence angle. blacksquare bigskip noindent textbfRemark. This is the operating principle of the corner reflector or corner cube retroreflector: any incoming ray striking all three faces is reflected straight back parallel to its source. This is why corner-cube reflectors are used on bicycle reflectors road signs and even the Apollo lunar retroreflector arrays used to measure the Earth--Moon distance.