Q. If $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}\$and$\ \left| \overrightarrow{\mathbf{a}} \right| = 5,\ \ \left| \overrightarrow{\mathbf{b}} \right| = 4\$and$\ \left| \overrightarrow{\mathbf{c}} \right| = 3,\$then the value of $|\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}}|$ is
Explanation
(c)
Given, $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$
$\therefore|\ \overrightarrow{\mathbf{a}}|^{2} + |\overrightarrow{\mathbf{b}}|^{2} + |\overrightarrow{\mathbf{c}}|^{2} + 2\left\lbrack \overrightarrow{\mathbf{a}} \bullet \ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right\rbrack = 0$
$\Longrightarrow 25 + 16 + 9 + 2\left\lbrack \overrightarrow{\mathbf{a}} \bullet \ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right\rbrack = 0$
$\Longrightarrow 2\left\lbrack \overrightarrow{\mathbf{a}} \bullet \ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right\rbrack = - 50$
$\Longrightarrow \left\lbrack \overrightarrow{\mathbf{a}} \bullet \ \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right\rbrack = - 25$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.