Question

If $\hat{a},\hat{b}$ and $\hat{c}$ are three unit vectors inclined to each other at an angle $\theta$, then the maximum value of $\theta$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\frac{2\pi }{3}$
• D. $\frac{5\pi }{3}$

Answer 1

The correct option is C $\frac{2\pi }{3}$

$(\hat{a}+\hat{b}+\hat{c}{)}^2\ge 0$
$3+2(\hat{a}\cdot \hat{b}+\overrightarrow{b}\cdot \hat{c}+\overrightarrow{c}\cdot \overrightarrow{a})\ge 0$
$3+6cos\theta \ge 0$
$cos\theta \ge -\frac{1}{2}$
$\Rightarrow$ maximum $\theta =\frac{2\pi }{3}$