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