Question

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 DRs of the normal to the plane of mirror are 1, –1, 1, then 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

Answer: (A)

$\frac{1}{3},\frac{2}{3},\frac{2}{3}$

Answer 2

Let the source of light be situated at A(a, 0, 0), where a ¹ 0. Let OA be the incident ray and OB the reflected ray. ON is the normal to the mirror at O

ŠAON = ŠNOB = $\frac{\theta}{2}$ (say)

DRs of OA are a, 0, 0 and so its DCs are 1, 0, 0

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

Let l, m, n be the DCs of the reflected ray OB.

Then $\frac{n + 0}{2\cos\frac{\theta}{2}}$ = $\frac{1}{\sqrt{3}}$

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

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

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