Q139 / 155 · WBJEE 2024 · Physics
If $\hat{n}_1, \hat{n}_2$ and $\hat{t}$ represent, unit vectors along the incident ray, reflected ray and normal to the surface respectively, then
- Correct answerA$\quad \hat{n}_2=\hat{n}_1-2\left(\hat{n}_1 \cdot \hat{t}\right) \hat{\mathrm{t}}$
- B$\hat{n}_2=\hat{n}_1+2\left(\hat{n}_1 \cdot \hat{t}\right) \hat{\mathrm{t}}$
- C$\hat{n}_2=-\hat{n}_1$
- D$\hat{n}_2=2 \hat{n}_1-\left(\hat{n}_1 \times \hat{t}\right) \cdot \hat{n}_1$
Solution
$\begin{aligned} & \text { Hint: } \hat{n}_1=-\cos \theta \hat{\mathrm{t}}+\sin \theta \hat{\mathrm{i}} \\ & \hat{\mathrm{n}}_2=\cos \theta \hat{\mathrm{t}}+\sin \theta \hat{\mathrm{i}} \\ & \hat{\mathrm{n}}_2-\hat{\mathrm{n}}_1=2 \cos \theta \hat{\mathrm{t}}=-2(\hat{\mathrm{n}} \cdot \hat{\mathrm{t}}) \hat{\mathrm{t}} \\ & \hat{\mathrm{n}}_1 \cdot \hat{\mathrm{t}}=-\cos \theta \\ & \hat{n}_2=\hat{n}_1-2\left(\hat{n}_1 \cdot \hat{\mathrm{t}}\right) \hat{\mathrm{t}}\end{aligned}$
Hence, the correct answer is option: 'A'.