Question

If the unit vectors $\overrightarrow{e_1}$ and $\overrightarrow{e_2}$ are inclined at an angle $2\theta$ and $\left|\overrightarrow{e_1}-\overrightarrow{e_2}\right|<1$, then for $\theta \in \left[0,\pi \right],\theta$ may be in the interval :

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

Answer 1

Answer: (A)

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

$\left|\overline{e_1}\right|=\left|\overline{e_2}\right|=1|\overline{e_1}-\overline{e_2}|<1{|\overline{e_1}-\overline{e_2}|}^2<1^2{\overline{e_1}}^2+{\overline{e_2}}^2-2\overline{e_1}\overline{e_2}\cos 2\theta <1$

$2-2\cos 2\theta <1$ as $\theta \in [0,\pi ]$

$2\cos 2\theta >1$ $2\theta \in [0,2\pi ]$

$\cos 2\theta >\frac{1}{2}$

$2\theta \in (0,\frac{\pi }{3})\cup (\frac{5\pi }{3},2\pi )\therefore \theta \in (0,\frac{\pi }{6})\cup (\frac{5\pi }{6},\pi )$