Question

If the unit vectors $a$ and $b$ are inclined at angle $2 \theta(0 \leq$ $\theta \leq \pi$ ) and $|a-b|<1$, then $\theta$ lies in the interval

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

Answer 1

Answer: (A)

$a \cdot b=|a||b| \cos 2 \theta$
$\Rightarrow a \cdot b=(1)(1) \cos 2 \theta=\cos 2 \theta .$
$|a-b|<1$
$\Rightarrow a^{2}+b^{2}-2 a \cdot b<1 $
$\Rightarrow 1+1-2 \cos \theta<1$
$\Rightarrow 2(1-\cos 2 \theta)<1 $
$\Rightarrow 2\left(2 \sin ^{2} \theta\right)<1$
$\Rightarrow \sin ^{2} \theta<\frac{1}{4}$
$ \Rightarrow \theta$ lies in $\left[0, \frac{\pi}{6}\right)$