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)

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