Q. If $\overrightarrow{\mathbf{x}}\mathbf{\bullet}\overrightarrow{\mathbf{a}}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{\bullet}\overrightarrow{\mathbf{c}}\mathbf{=}0$ where $\overrightarrow{\mathbf{x}}$ is a non-zero vector. Then, $\lbrack\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}}\mathbf{\ \ }\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}\rbrack$ is equal to
Explanation
(d)
$\because\overrightarrow{\mathbf{x}}\mathbf{\bullet}\overrightarrow{\mathbf{a}}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{\bullet}\overrightarrow{\mathbf{b}}\mathbf{=}\overrightarrow{\mathbf{x}}\mathbf{\bullet}\overrightarrow{\mathbf{c}} = 0$
For non-zero vector $\overrightarrow{\mathbf{x}}$
$\left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack = 0$ (three vectors $\overrightarrow{\mathbf{a}}\mathbf{,\ }\overrightarrow{\mathbf{b}}\mathbf{,\ }\overrightarrow{\mathbf{c}}$ are coplanar )
and $\lbrack\ \overrightarrow{\mathbf{a}}\mathbf{\times \ }\overrightarrow{\mathbf{b}}\mathbf{\ \ }\overrightarrow{\mathbf{b}}\mathbf{\ \times}\overrightarrow{\mathbf{c}}$ $\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}\ \rbrack\ \$
$= \left\lbrack \ \overrightarrow{\mathbf{a}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{c}}\ \right\rbrack^{2} = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.