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

Question

Dr. Abubacker · Accepted Answer

(a) Unit vector along incident ray
$\mathbf{Î} = \frac{\left( 2\widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} + 4\widehat{\mathbf{k}} ight)}{\sqrt{29}}$
Unit vector along normal $\widehat{\mathbf{N}} = \frac{\left( 3\widehat{\mathbf{i}}\  - \ 6\widehat{\mathbf{j}}\  + 2\widehat{\mathbf{k}} ight)}{7}$
Unit vector along reflected ray
$\widehat{\mathbf{R}}\mathbf{=}\widehat{\mathbf{i}} - 2\left( \widehat{\mathbf{i}}\mathbf{\ .\ }\widehat{\mathbf{j}} ight)\widehat{\mathbf{N}}$
$\mathbf{\Rightarrow \ }\widehat{\mathbf{R}}\mathbf{=}\frac{\left( 2\widehat{\mathbf{i}} - 3\widehat{\mathbf{j}} + 4\widehat{\mathbf{k}} ight)}{\sqrt{29}} - 2\left( \frac{32}{7\sqrt{29}} ight) 	imes \left\lbrack \frac{\left( 3\widehat{\mathbf{i}}\  - \ 6\widehat{\mathbf{j}}\  + 2\widehat{\mathbf{k}} ight)}{7} ightbrack$
$\mathbf{\Rightarrow}\widehat{\mathbf{R}}\mathbf{=}\frac{\left( - 94\widehat{\mathbf{i}} + 237\widehat{\mathbf{j}} + 68\widehat{\mathbf{k}} ight)}{49\sqrt{29}}$

Explanation

Reviewed by Dr. Abubacker

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