Question

A plane mirror is placed at the origin so that the direction ratios of its normal are $(1,-1,1)$. A ray of light, coming along the positive direction of the x-axis, strikes the mirror. The direction cosines of the reflected ray are

• A. $\frac{1}{3},\frac{2}{3},\frac{2}{3}$
• B. $-\frac{1}{3},\frac{2}{3},\frac{2}{3}$
• C. $-\frac{1}{3},-\frac{2}{3},-\frac{2}{3}$
• D. $-\frac{1}{3},-\frac{2}{3},\frac{2}{3}$

Answer 1

The correct option is D $-\frac{1}{3},-\frac{2}{3},\frac{2}{3}$

Let the source of light be situated at A (a,0,0) where a is not equal to 0.

let OA be the incident ray and OB the reflected ray

ON is the normal to the mirror at O

D.R's of OA are $0,0,1$

and its D.R's are $1,0,0$

D.R's of ON are $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

therefore cos $\frac{\theta }{2}$ = $\frac{1}{\sqrt{3}}$

let l,m,n be the D.R's of the reflected ray OA then,

$l$ =$\frac{2}{3}-1,$ $m$= $\frac{-2}{3}$, $n$ = $\frac{2}{3}$

l= $\frac{-1}{3}$, m = $\frac{-2}{3},$ n = $\frac{2}{3}$

Hence the D.R's of the reflected ray are $\frac{-1}{3}$, $\frac{-2}{3},$, $\frac{2}{3}$.