A mirror and source of light are kept at the origin at a point on the positive x-axis respectively a ray of light from the source strikes the mirror and is reflected. If $(1,-1,1)$ are the d.r’s of a normal to the plane then the d.c’s of reflected ray are

Let the source of light situated at A (a, 0, 0), Where  $\mathrm{a}\ne 0$
Let OA be the normal to the mirror at 0.
$\therefore \mathrm{\angle }\mathrm{NOA}=⌊\mathrm{NOB}=\frac{\mathrm{\theta }}{2}$

D.r’s of OA are (a, 0, 0) and d.c’s are (1, 0, 0)
$\therefore$ d.c’s of ON are  $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

$\therefore \cos \mathrm{\theta }=\frac{1}{\sqrt{3}}$
Let l,m,n be the d.c’s of the reflected ray OB, then as in the problem.
$\begin{array}{l}\Rightarrow \frac{\mathrm{l}+1}{\frac{2\cos \mathrm{\theta }}{2}}=\frac{1}{\sqrt{3}},\frac{\mathrm{m}+0}{\frac{2\cos \mathrm{\theta }}{2}}=-\frac{1}{\sqrt{3}}\text{ and }\frac{\mathrm{n}+0}{\frac{2\cos \mathrm{\theta }}{2}}=\frac{1}{\sqrt{3}}\\ \mathrm{l}=\frac{-1}{\sqrt{3}},\mathrm{m}=-\frac{2}{3},\mathrm{n}=\frac{2}{3}\end{array}$

Hence d.c’s of the reflected ray are $\frac{-1}{3},\frac{-2}{3},\frac{2}{3}$