First slide

Reflection of light

Question

A ray of light is incident on a plane mirror, along the direction given by, $\vec{A} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ . Find the unit vector along the reflected ray. Take normal to mirror along the direction of $\vec{B} = 3 \hat{i} - 6 \hat{j} + 2 \hat{k}$

Moderate

A

$\frac{- 94 \hat{i} + 237 \hat{j} + 68 \hat{k}}{49 \sqrt{29}}$
B

$\frac{- 94 \hat{i} + 68 \hat{j} - 237 \hat{k}}{49 \sqrt{29}}$
C

$\frac{3 \hat{i} + 6 \hat{j} - 2 \hat{k}}{7}$
D

$N o n e o f t h e s e$

Solution

Unit vector along reflected ray,
$\hat{e_{R}} = \hat{e_{I}} - 2 (\hat{n} . \hat{e_{I}}) \hat{n}$
$= \frac{2 \hat{i} - 3 \hat{j} + 4 \hat{k}}{\sqrt{29}} - 2 ((\frac{3 \hat{i} - 6 \hat{j} + 2 \hat{k}}{7}) . (\frac{2 \hat{i} - 3 \hat{j} + 4 \hat{k}}{\sqrt{29}})) (\frac{3 \hat{i} - 6 \hat{j} + 2 \hat{k}}{7})$
$= \frac{2 \hat{i} - 3 \hat{j} + 4 \hat{k}}{\sqrt{29}} - 2 [\frac{6 + 18 + 8}{7 \times \sqrt{29} \times 7}] (3 \hat{i} - 6 \hat{j} + 2 \hat{k}) =$ $\frac{- 94 \hat{i} + 237 \hat{j} + 68 \hat{k}}{49 \sqrt{29}}$

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