Question

If $a$ and $b$ are unit vectors and $\theta$ is the angle between them, then $|a+b|<1$, if

• A. $\theta =\frac{\pi }{2}$
• B. $\theta <\frac{\pi }{3}$
• C. $\pi \ge \theta >\frac{2\pi }{3}$
• D. $\frac{\pi }{3}<\theta <\frac{2\pi }{3}$

Answer 1

The correct option is D $\frac{\pi }{3}<\theta <\frac{2\pi }{3}$

Given $|\overrightarrow{a}|=1$ and $|\overrightarrow{b}|=1$
$|\overrightarrow{a}+\overrightarrow{b}|<1$
$\Rightarrow |\overrightarrow{a}+\overrightarrow{b}{|}^2<1$
$\Rightarrow |\overrightarrow{a}{|}^2+|\overrightarrow{b}{|}^2+2|\overrightarrow{a}||\overrightarrow{b}|\cos \theta <1$
$\Rightarrow 1+1+2\cos \theta <1$
$\Rightarrow \cos \theta <-\frac{1}{2}$
$\therefore -1\le \cos \theta <-\frac{1}{2}$
$\Rightarrow \pi \ge \theta >\frac{2\pi }{3}$