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

Answer 1

The correct option is D

$\overrightarrow{b}=\overrightarrow{a}-2\left(\overrightarrow{a·}\overrightarrow{c}\right)\overrightarrow{c}$

The explanation for the correct option:

In the case of option D,

1. The incident ray, reflected ray, and the outward drawn normal are denoted by the unit vectors$\overrightarrow{a},\overrightarrow{b}$ and$\overrightarrow{c}$.
2. From the law of refraction, $\angle i=\angle r$ let $\angle i=\angle r=\theta$
3. 
4. From figure, $$\begin{gathered} \overrightarrow{a}=\sin \theta \hat{i}-\cos \theta \hat{j} \\ \overrightarrow{b}=\sin \theta \hat{i}+\cos \theta \hat{j} \\ \overrightarrow{c}=\hat{j} \end{gathered}$$
5. Now take option D

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

Therefore the correct relation is $\overrightarrow{b}=\overrightarrow{a}-2\left(\overrightarrow{a·}\overrightarrow{c}\right)\overrightarrow{c}$

The explanation for the incorrect option:

In the case of option A,

1. Take option A, $\overrightarrow{b}=2\overrightarrow{a}+\overrightarrow{c}$
2. On further solving$$\begin{gathered} =2\overrightarrow{a}+\overrightarrow{c} \\ =2\left(\sin \theta \hat{i}-\cos \theta \hat{j}\right)\hat{j} \\ =2\left(-\cos \theta \right) \\ \ne \overrightarrow{b} \end{gathered}$$
3. Not correct relation

In the case of option B,

1. Take $\overrightarrow{b}=\overrightarrow{a}-\overrightarrow{c}$
2. On further solving, $$\begin{gathered} =\overrightarrow{a}-\overrightarrow{c} \\ =\sin \theta \hat{i}-\cos \theta \hat{j}-\hat{j} \\ =\sin \theta -\left(1+\cos \theta \right)\hat{j} \\ \ne \overrightarrow{b} \end{gathered}$$
3. Not correct relation.

In the case of option C,

1. Similarly, take $\overrightarrow{b}=\overrightarrow{a}+2\overrightarrow{c}$
2. On Further solving, $$\begin{gathered} =\overrightarrow{a}+2\overrightarrow{c} \\ =\left(\sin \theta \hat{i}-\cos \theta \hat{j}\right)+2\left(\hat{j}\right) \\ =\sin \theta \hat{i}-\hat{j}\left(\cos \theta -2\right) \\ \ne \overrightarrow{b} \end{gathered}$$
3. Not correct relation.

Hence, option D is the correct answer.