Question

If the unit vectors $a$ and $b$ are inclined at angle $2\theta (0\le$ $\theta \le \pi$ ) 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)

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