A mirror and a source  of light are situated at the origin O and at a point on OX, respectively. A ray of light from the
source strikes the mirror and is reflected. If the direction ratios of the normal to the plane of mirror are 1, -1, 1,
then direction cosines for the reflected ray arc

$\text{ Let the source of light be situated at }A(a,0,0)\text{, where }a\ne 0\text{. Let }OA\text{ be the incident ray and }OB\text{ the reflected ray. }$

$ON\text{ is the normal to the mirror at }O\text{. }$

$\mathrm{\angle }\mathrm{AON}=\mathrm{\angle }NOB=\frac{\theta }{2}\text{ (say) }$

$\text{ DRs of }OA\text{ are }a,0,0\text{ and so its DCs are }1,0,0\text{. }$

$\text{ DCs of }ON\text{ are }\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\therefore \cos \frac{\theta }{2}=\frac{1}{\sqrt{3}}$

$\text{ Let }l,m,n\text{ be the DCs of the reflected ray }OB\text{. Then }$

$\begin{array}{l}\frac{l+1}{2\cos \frac{\theta }{2}}=-\frac{m+0}{2\cos \frac{\theta }{2}}=\frac{n+0}{2\cos \frac{\theta }{2}}=\frac{1}{\sqrt{3}}\\ \Rightarrow l=\frac{2}{3}-1,m=\frac{-2}{3},n=\frac{2}{3}\\ \Rightarrow l=-\frac{1}{3},m=-\frac{2}{3},n=\frac{2}{3}\end{array}$

$\text{ Hence, DCs of the reflected ray, are }-\frac{1}{3},-\frac{2}{3},\frac{2}{3}\text{. }$