Question

${\hat{n}}_1$ is the unit vector along incident ray, ${\hat{n}}_2$ along the normal to plane mirror and $\widehat{n_3}$ is the unit vector along reflected ray direction, then which of the following must be true?

• A. ${\hat{n}}_1.{\hat{n}}_2=0$
• B. ${\hat{n}}_1.{\hat{n}}_3=0$
• C. $({\hat{n}}_1\times {\hat{n}}_2).{\hat{n}}_3=0$
• D. $({\hat{n}}_1\times {\hat{n}}_2)\times {\hat{n}}_3=0$

Answer 1

The correct option is C $({\hat{n}}_1\times {\hat{n}}_2).{\hat{n}}_3=0$

From the laws of reflection
the incident ray, reflected ray and normal lie on the same plane.


Here ${\hat{n}}_1,{\hat{n}}_{2}and{\hat{n}}_3$ are in the same $x-y$ plane.
From rule of cross product of vector
$({\hat{n}}_1\times {\hat{n}}_2)$ will give an unit vector which is perpendicular to the plane of ${\hat{n}}_1,{\hat{n}}_2$ i.e. $\perp$ to the (x-y) plane.
$\because ({\hat{n}}_1\times {\hat{n}}_2)$ $\perp$ ${\hat{n}}_3$
$\Rightarrow ({\hat{n}}_1\times {\hat{n}}_2)\cdot {\hat{n}}_3=0$ (from dot product)
Hence, option $\left(c\right)$ is the correct answer.