Question

${\hat{n}}_1$ is the unit vector along incident ray, ${\hat{n}}_2$ along normal and ${\hat{n}}_3$ is the unit vector along reflected ray, then which of the following must be true?

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

Answer 1

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

From the figure let us consider the options.

(A) ${\hat{n}}_1.{\hat{n}}_2=\left|n_1\right|\left|n_2\right|\cos \theta \ne 0$

(B) ${\hat{n}}_1.{\hat{n}}_3=\left|n_1\right|\left|n_3\right|\cos 2\theta \ne 0$

(C) $\left({\hat{n}}_1\times {\hat{n}}_2\right).{\hat{n}}_3=\left|{\hat{n}}_1\times {\hat{n}}_2\right|\left|n_3\right|\cos {90}^{\circ }=0$

(D) $\left({\hat{n}}_1\times {\hat{n}}_2\right)\times {\hat{n}}_3=\left|{\hat{n}}_1\times {\hat{n}}_2\right|\left|n_3\right|\sin {90}^{\circ }\ne 0$

Hence the correct answer is option (C).