Question

The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors, $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ respectively. Then, choose the correct relation for these vectors.

• A. $\overrightarrow{b}=\overrightarrow{2a}+\overrightarrow{c}$
• B. $\overrightarrow{b}=\overrightarrow{a}-2(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{c}$
• C. $\overrightarrow{b}=\overrightarrow{a}+2\overrightarrow{c}$
• D. $\overrightarrow{b}=\overrightarrow{a}-\overrightarrow{c}$

Answer 1

The correct option is B $\overrightarrow{b}=\overrightarrow{a}-2(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{c}$


From diagram,

$\overrightarrow{a}=\sin \theta \hat{i}-\cos \theta \hat{j}$

$\overrightarrow{b}=\sin \theta \hat{i}+\cos \theta \hat{j}$

$\overrightarrow{c}=\hat{j}$

So,
$\overrightarrow{a}-2(\overrightarrow{a}.\overrightarrow{c})\overrightarrow{c}$
$=(\sin \theta \hat{i}-\cos \theta \hat{j})-2\left[\right(\sin \theta \hat{i}-\cos \theta \hat{j}).(\hat{j}\left)\right]\hat{j}$$=\sin \theta \hat{i}-\cos \theta \hat{j}+2\cos \theta \hat{j}$$=\sin \theta \hat{i}+\cos \theta \hat{j}=\overrightarrow{b}$

Hence, option $\left(D\right)$ is correct.