Question

If the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are inclined at an angle 2 $\theta$ such that $|\overrightarrow{a}-\overrightarrow{b}|$ <1 and 0 $\le \theta \le \pi$, then $\theta$ lies in the interval

• A. $[0,\frac{\pi }{6})$
• B. $(\frac{5\pi }{6},\pi ]$
• C. $[\frac{\pi }{6},\frac{\pi }{2}]$
• D. $[\frac{\pi }{2},\frac{\pi }{6}]$

Answer 1

Answer: (A), (B)

The correct options are
A $[0,\frac{\pi }{6})$
B $(\frac{5\pi }{6},\pi ]$

We have, $|\overrightarrow{a}-\overrightarrow{b}{|}^2=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2(\overrightarrow{a}.\overrightarrow{b})$
$\Rightarrow |\overrightarrow{a}-\overrightarrow{b}{|}^2=|\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2-2|\overrightarrow{a}||\overrightarrow{b}|cos2\theta \Rightarrow |\overrightarrow{a}-\overrightarrow{b}{|}^2=2-2cos2\theta [\because |\overrightarrow{a}|=|\overrightarrow{b}|=1]\Rightarrow |\overrightarrow{a}-\overrightarrow{b}|=2|sin\theta |$
Now, $|\overrightarrow{a}-\overrightarrow{b}|<1\Rightarrow 2|sin\theta |<1$
$\Rightarrow |sin\theta |<\frac{1}{2}\Rightarrow \theta \in [0,\frac{\pi }{6})or\theta \in (\frac{5\pi }{6},\pi ]$