Question

A mirror and a source of light are situated at the origin O and a point on OX(x-axis) respectively. A ray of light from the source strikes the mirror and is reflected. If the DRs of the normal to the plane of mirror are 1, – 1, 1, the DCs for 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 $a\ne 0$. Let OA be the incident ray and OB the reflected ray ON is the normal to the mirror at O

$\angle AON=\angle NOB=\frac{\theta }{2}\left(say\right)$
DR’s of OA are a, 0, 0 and so its DC's are 1, 0, 0
DC’s of ON are $\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},$
$\therefore cos\frac{0}{2}=\frac{1}{\sqrt{3}}$
Let l, m, n be the DC’s of the reflected ray OB.
$\Rightarrow l=\frac{2}{3}-1,m=-\frac{2}{3},n=\frac{2}{3}\Rightarrow l=-\frac{1}{3},m=-\frac{2}{3},n=\frac{2}{3}$
Hence, DC’s of the reflected ray are
$-\frac{1}{3},-\frac{2}{3},\frac{2}{3}$