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