Q. If $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{0}}$ and $\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 0$, then
A) $\overrightarrow{\mathbf{a}}\bot\overrightarrow{\mathbf{b}}$
B) $\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}$
C) $\overrightarrow{\mathbf{a}} = \overrightarrow{\mathbf{0}}\ and\ \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{0}}$
D) $\overrightarrow{\mathbf{a}} = \overrightarrow{\mathbf{0}}\ or\ \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{0}}$
Explanation
(d)
Given, $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{0}}\ and\overrightarrow{\mathbf{a}} \bullet \overrightarrow{\mathbf{b}} = 0$
$\Longrightarrow \ \overrightarrow{\mathbf{a}}$ is parallel to $\overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{a}}$ is perpendicular to $\overrightarrow{\mathbf{b}}$ which is possible only if
$\overrightarrow{\mathbf{a}} = \overrightarrow{\mathbf{0}}\ \ or\ \overrightarrow{\mathbf{b}} = \overrightarrow{\mathbf{0}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.