Question

If two unit vectors $\overline{a}$ and $\overline{b}$ are inclined at an angle $2\theta (0<\theta <\pi )$ and $|\overline{a}-\overline{b}|\le 1$, then $\theta \in$

• A. $[0,\frac{\pi }{6})$
• B. $[0,\frac{\pi }{3}\left]\cup \right[\frac{2\pi }{3},\pi ]$
• C. $[0,\frac{\pi }{6}]\cup [\frac{5\pi }{6},\pi ]$
• D. $\varphi$

Answer 1

Answer: (A)

$[0,\frac{\pi }{6})$

$|\overline{a}-\overline{b}|=\sqrt{|\overline{a}{|}^2+|\overline{b}{|}^2-2|\overline{a}||\overline{b}|\cos \theta }$
$=\sqrt{2(1-\cos \theta )}$
$|\overline{a}-\overline{b}|=2\sin \theta$

By the condition
$2\sin \theta \le 1$
$\sin \theta \le \frac{1}{2}$
$\theta ϵ[0,\frac{\pi }{6}]$