Question

Which of the following relations is true for two unit vectors $\hat{ A }$ and $\hat{ B }$ making an angle $\theta$ to each other?

• 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. $|\hat{ A }-\hat{ B }|=|\hat{ A }+\hat{ B }| \cos \frac{\theta}{2}$

Answer 1

Answer: (B)

$|\hat{ A }+\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}+2|\hat{ A } \| \hat{ B }| \cos \theta}$
$=\sqrt{1+1+2 \cos \theta}$
$=\sqrt{2(1+\cos \theta)}$
$=\sqrt{2 \times 2 \cos ^{2} \frac{\theta}{2}}$
$=2 \cos \frac{\theta}{2}$
$|\hat{ A }-\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}-2|\hat{ A }||\hat{ B }| \cos \theta}$
$=\sqrt{2-2 \cos \theta}$
$=2 \sin \frac{0}{2}$
$\frac{|\hat{ A }+\hat{ B }|}{|\hat{ A }-\hat{ B }|}=\cot \frac{\theta}{2}$