એકબીજા સાથે $\theta$ કોણ બનાવતા બે એકમ સદિશો $\hat{A}$ અને $\hat{B}$ માટે નીચેનામાંથી કયો સંબંધ સાચો છે $?$

Question

A$|\hat{ A }+\hat{ B }|=|\hat{ A }-\hat{ B }| \tan \frac{\theta}{2}$
B$|\hat{ A }-\hat{ B }|=|\hat{ A }+\hat{ B }| \tan \frac{\theta}{2}$
C$|\hat{ A }+\hat{ B }|=|\hat{ A }-\hat{ B }| \cos \frac{\theta}{2}$
D$|\overrightarrow{ A }-\hat{ B }|=|\overrightarrow{ A }+\hat{ B }| \cos \frac{\theta}{2}$

JEE MAIN 2022, Diffcult

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Solution

b
$|\hat{ A }+\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}+2|\hat{ A } \| \hat{ B }| \cos \theta}$

$=\sqrt{1+1+2 \cos \theta}$

$=\sqrt{2(1+\cos \theta)}$

$=\sqrt{2 \times 2 \cos ^{2} \frac{\theta}{2}}$

$=2 \cos \frac{\theta}{2}$

$|\hat{ A }-\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}-2|\hat{ A }||\hat{ B }| \cos \theta}$

$=\sqrt{2-2 \cos \theta}$

$=2 \sin \frac{\theta}{2}$

$\frac{|\hat{ A }+\hat{ B }|}{|\hat{ A }-\hat{ B }|}=\cot \frac{\theta}{2}$

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