Q. IF $\overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) = (\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \times \overrightarrow{\mathbf{c}}$, where $\overrightarrow{\mathbf{a}}$, $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are any three vectors such that $\overrightarrow{\mathbf{a}}.\overrightarrow{\mathbf{b}} \neq 0,\ \ \ \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} \neq 0,$ then $\overrightarrow{\mathbf{a}}\ and\ \overrightarrow{\mathbf{c}}$ are
Explanation
(c)
$\because\ \left( \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right) \times \overrightarrow{\mathbf{c}} = \overrightarrow{\mathbf{a}} \times \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right)$
$\Longrightarrow \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} \right)\overrightarrow{\mathbf{b}} - \left( \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} \right)\overrightarrow{\mathbf{a}} = \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{c}} \right)\overrightarrow{\mathbf{b}} - \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{c}}$
$\Longrightarrow \left( \overrightarrow{\mathbf{b}} \bullet \overrightarrow{\mathbf{c}} \right)\overrightarrow{\mathbf{a}} = \left( \overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{c}}$
$\Longrightarrow \ \overrightarrow{\mathbf{a}}$ is parallel to $\overrightarrow{\mathbf{c}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.