Question

Unit vectors $\hat{a}$ and $\hat{b}$ are inclined at an angle $2\theta$ and $|\hat{a}-\hat{b}|<1$. If $0\le \theta <\pi$, then $\theta$ may belong to.

• A. $[0,\frac{\pi }{6})$
• B. $(\frac{\pi }{6},\frac{5\pi }{6})$
• C. $[\frac{\pi }{6},\frac{\pi }{2}]$
• D. None of these

Answer 1

The correct option is A $[0,\frac{\pi }{6})$

Given vectors

$\overrightarrow{a}$ and $\overrightarrow{b}$

$|\overrightarrow{a}-\overrightarrow{b}|<1$

Since both sides are positive, hence squarring

$\left|{\overrightarrow{a}}^2\right|+\left|{\overrightarrow{b}}^2\right|-2.\left|\overrightarrow{a}\right|.\left|\overrightarrow{b}\right|.\cos \left(2\theta \right)<1$

$1+1-2\cos \left(2\theta \right)<1$

$2(1-\cos \left(2\theta \right))<1$

$1-\cos \left(2\theta \right)<\frac{1}{2}$

$\cos \left(2\theta \right)>\frac{1}{2}$

$2\theta \in [0,\frac{\pi }{3})$ ...... $[\because \theta \in [0,\pi \left)\right]$

$\theta \in [0,\frac{\pi }{6})$