Q. If $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$ and $\left| \overrightarrow{\mathbf{a}} \right| = 3,\left| \overrightarrow{\mathbf{b}} \right| = 5,\overrightarrow{\mathbf{a}} + \left| \overrightarrow{\mathbf{c}} \right| = 7,$ then angle between $\overrightarrow{\mathbf{a}}\ and\ \overrightarrow{\mathbf{b}}$ is
A) $\frac{\pi}{6}$
B) $\frac{\pi}{3}$
C) $\frac{\pi}{2}$
D) $\pi$
Explanation
(b)
Given $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} = - \overrightarrow{\mathbf{c}}$
$\Longrightarrow \ |\overrightarrow{\mathbf{a}}|^{2} + |\overrightarrow{\mathbf{b}}|^{2} + 2\ \left| \overrightarrow{\mathbf{a}} \right||\overrightarrow{\mathbf{b}}|cos\ \theta = = |\overrightarrow{\mathbf{c}}|^{2}$
$\Longrightarrow 9 + 25 + 2 \bullet 3 \bullet 5\cos{\theta = 49}$
$\Longrightarrow \cos{\theta = \frac{1}{2}}$
$\Longrightarrow \ \theta = \frac{\pi}{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.