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Physics Question on Ray optics and optical instruments

A ray is incident on a plane surface. If i^+j^−k^\hat{i} + \hat{j} - \hat{k} represents a vector along the direction of incident ray.i^+j^\hat{i} + \hat{j} is a vector along normal on incident point in the plane of incident and reflected ray, then vector along the direction of reflected ray is

A

−119(−3i^+3j^+k^)-\frac{1}{\sqrt{19}} (-3\hat{i} + 3\hat{j} + \hat{k})

B

119(3i^+3j^−k^)\frac{1}{\sqrt{19}} (3\hat{i} + 3\hat{j} - \hat{k})

C

−13(i^+j^+k^)-\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})

D

k^\hat{k}

Answer

−13(i^+j^+k^)-\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})

Explanation

Solution

According to law of reflection in vector form,
n2^=n1^−2(n1^.n^)n^\hat{n_2} = \hat{n_1} - 2 (\hat{n_1} .\hat{n}) \hat{n}
Here, n1^\hat{n_1} = the unit vector along incident ray
=i^+j^−k^3= \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}}
n^\hat{n} = unit vector along normal on incident point
=i^+j^2= \frac{\hat{i} + \hat{j}}{\sqrt{2}}
n2^\hat{n_2} =unit vector along the direction of reflected ray
Using the formula, we get
\hat{n_2} = \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}} - 2 \bigg\\{ \bigg( \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}}\bigg) . \bigg( \frac{\hat{i} + \hat{j}}{\sqrt{2}} \bigg). \bigg( \frac{\hat{i} + \hat{j} }{\sqrt{2}} \bigg) \bigg\\}
= \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}} - 2 \bigg\\{ \frac{1}{\sqrt{6}} + \frac{1}{\sqrt{6}} \bigg\\} \frac{\hat{i} + \hat{j}}{\sqrt{2}}
=i^+j^−k^3−46(i^+j^2)= \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}} - \frac{4}{\sqrt{6}} \bigg( \frac{\hat{i} + \hat{j}}{\sqrt{2}} \bigg)
=i^+j^−k^3−4i^12−4j^12= \frac{\hat{i} + \hat{j} - \hat{k}}{\sqrt{3}} - \frac{4\hat{i}}{\sqrt{12}}- \frac{4\hat{j}}{\sqrt{12}}
=2i^+2j^−2k^−4i^−4j^23= \frac{2 \hat{i} +2 \hat{j} - 2\hat{k} -4\hat{i} - 4\hat{j} }{2\sqrt{3}} (∵ 12=23)(\because \, \sqrt{12} = 2 \sqrt{3} )
=−2i^−2j^−2k^23=−i^−j^−k^3= \frac{-2 \hat{i} -2 \hat{j} - 2\hat{k} }{2\sqrt{3}} = \frac{- \hat{i} - \hat{j} -\hat{k}}{\sqrt{3}}
=−13(i^+j^+k^)= - \frac{1}{\sqrt{3}} ( \hat{i} + \hat{j} +\hat{k})