Q. Let $\overrightarrow{\mathbf{a}},\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}\$ be vectors with magnitude 3,4 and 5 respectively and $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{0}}$ , 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
(d)
Given $\overrightarrow{\mathbf{a}} + \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{0}}$
$\Longrightarrow |\overrightarrow{\mathbf{a}}|^{2} + |\overrightarrow{\mathbf{b}}|^{2} + |\overrightarrow{\mathbf{c}}|^{2} + 2\left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right) = 0$
$\Longrightarrow \ 3^{2} + 4^{2} + 5^{2} + 2\left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} \right) = 0$
$\Longrightarrow \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} + \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} + \overrightarrow{\mathbf{c}} \bullet \overrightarrow{\mathbf{a}} = - 25$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.