Q. $\overrightarrow{\mathbf{a}}\mathbf{\times \lbrack}\overrightarrow{\mathbf{a}}\mathbf{\times}\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)\rbrack$ is equal to

A) $\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)\mathbf{\bullet}\left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)$
B) $\overrightarrow{\mathbf{a}}\mathbf{\bullet}\left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)\mathbf{-}\overrightarrow{\mathbf{b}}\mathbf{\bullet}\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right)$
C) $\left\lbrack \overrightarrow{\mathbf{a}}\mathbf{\bullet}\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right) \right\rbrack\overrightarrow{\mathbf{\ a}}$
D) $\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} \right)\left( \mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)$
Explanation

(d)
$\overrightarrow{\mathbf{a}}\mathbf{\times}\left\lbrack \overrightarrow{\mathbf{a}}\mathbf{\times}\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}} \right) \right\rbrack\mathbf{=}\overrightarrow{\mathbf{a}}\mathbf{\times}\left\{ \mathbf{(\ }\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{a}}\mathbf{-}\left( \mathbf{\ }\overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\overrightarrow{\mathbf{b}}\ \}$ (Expanding by vector triple product)
$= \left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} \right)\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)\mathbf{-}\left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} \right)\mathbf{(}\overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{b}})$
$= \left( \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} \right)\left( \overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right)$ $\mathbf{\ (\because\ }\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\overrightarrow{\mathbf{a}} \right) = 0)$

Dr. Abubacker
Reviewed by Dr. Abubacker

Physics Educator & NEET/JEE Specialist

Ph.D. in Physics with 19+ years of teaching experience.

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