A ray is reflected in turn by three plain mirrors mutually at right angles to each other. The angle between the incident and the reflected rays is 

Ray after reflection from three mutually perpendicular mirrors becomes anti-parallel.   

$\begin{array}{rcl}\mathrm{\delta }& =& {\mathrm{\delta }}_1+{\mathrm{\delta }}_2+{\mathrm{\delta }}_3\\ & =& \left({180}^0-2\mathrm{\theta }\right)+\left({180}^0-2\left({90}^0-\mathrm{\theta }\right)\right)+0^0\\ & =& {180}^0\end{array}$

The ray never touch the third plane.