- Home
- Standard 11
- Physics
3-1.Vectors
hard
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$|\hat{ A }-\hat{ B }|=|\hat{ A }+\hat{ B }| \cos \frac{\theta}{2}$
(JEE MAIN-2022)
Solution
$|\hat{ A }+\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}+2|\hat{ A } \| \hat{ B }| \cos \theta}$
$=\sqrt{1+1+2 \cos \theta}$
$=\sqrt{2(1+\cos \theta)}$
$=\sqrt{2 \times 2 \cos ^{2} \frac{\theta}{2}}$
$=2 \cos \frac{\theta}{2}$
$|\hat{ A }-\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}-2|\hat{ A }||\hat{ B }| \cos \theta}$
$=\sqrt{2-2 \cos \theta}$
$=2 \sin \frac{\theta}{2}$
$\frac{|\hat{ A }+\hat{ B }|}{|\hat{ A }-\hat{ B }|}=\cot \frac{\theta}{2}$
$=\sqrt{1+1+2 \cos \theta}$
$=\sqrt{2(1+\cos \theta)}$
$=\sqrt{2 \times 2 \cos ^{2} \frac{\theta}{2}}$
$=2 \cos \frac{\theta}{2}$
$|\hat{ A }-\hat{ B }|=\sqrt{|\hat{ A }|^{2}+|\hat{ B }|^{2}-2|\hat{ A }||\hat{ B }| \cos \theta}$
$=\sqrt{2-2 \cos \theta}$
$=2 \sin \frac{\theta}{2}$
$\frac{|\hat{ A }+\hat{ B }|}{|\hat{ A }-\hat{ B }|}=\cot \frac{\theta}{2}$
Standard 11
Physics
