Q. 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 are 1, -1, 1, then direction cosines of the reflected rays are
Three Dimensional Geometry
Solution:
Let the ray of light comes along x-axis and strikes the mirror at the origin.
Direction cosines of normal are
$\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$ so. $cos \frac{\theta}{2} = \frac{1}{\sqrt{3}}$
Let the reflected ray has direction cosines l, m, n then
$\frac{l +1}{2\,cos \frac{\theta}{2}} = -\frac{1}{\sqrt{3}} \Rightarrow l = \frac{2}{3} -1 = -\frac{1}{3}$
$\frac{m +0}{2\,cos \frac{\theta }{2}} = -\frac{1}{\sqrt{3}} \Rightarrow m =- \frac{2}{3} \frac{n +0}{2\,cos \frac{\theta }{2}} = \frac{1}{\sqrt{3}} \Rightarrow n = \frac{2}{3}$
