Question

If $\vec{a}$ and $\vec{b}$ are unit vectors enclosing an angle $\theta$ and $\left|\vec{a}+\vec{b}\right|<1$, then

• 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

Answer: (C)

$\left|\vec{a}+\vec{b}\right|<1$
$\Rightarrow {\left|\vec{a}+\vec{b}\right|}^2<1$
$\Rightarrow {\left|\vec{a}\right|}^2+{\left|\vec{b}\right|}^2+2\vec{a}\cdot \vec{b}<1$
$\Rightarrow \vec{a}\cdot \vec{b}<-\frac{1}{2}$
$\Rightarrow \left|\vec{a}\right|\left|\vec{b}\right|cos\theta <-\frac{1}{2}$
$\Rightarrow 1\times 1\times cos\theta <-\frac{1}{2}$
$\Rightarrow cos\theta <-\frac{1}{2}$
$\Rightarrow -1\le cos\theta <-\frac{1}{2}$
$\Rightarrow \pi \ge \theta >\frac{2\pi }{3}$