Question

A ray of light is incident on a plane mirror along a vector $(\hat{i}+\hat{j}-\hat{k}).$ The normal on incident point is along $(\hat{i}+\hat{j}).$ The unit vector along the reflected ray is

• A. $\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$
• B. $-\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$
• C. $\frac{1}{\sqrt{3}}(-\hat{i}-\hat{j}+\hat{k})$
• D. $\frac{(\hat{i}-\hat{j}-\hat{k})}{\sqrt{3}}$

Answer 1

The correct option is B $-\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

Due to reflection, the component of the incident ray along normal is reversed and the perpendicular component is not changed.

Incident ray $=(\hat{i}+\hat{j})+(-\hat{k}).$

Here, $(\hat{i}+\hat{j})$ is reversed and $(-\hat{k})$ remains the same.

Hence, the reflected ray is
$\overrightarrow{r}=-\hat{i}-\hat{j}-\hat{k}$

Unit vector along the reflected ray is
$\hat{r}=\frac{\overrightarrow{r}}{|\overrightarrow{r}|}=\frac{-\hat{i}-\hat{j}-\hat{k}}{\sqrt{3}}=-\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

Hence, option $\left(b\right)$ is the correct answer.