Q. If $\overrightarrow{\mathbf{a}}\mathbf{,}\overrightarrow{\mathbf{b}}\mathbf{,}\overrightarrow{\mathbf{c}}$ are the three vectors mutually perpendicular to each other to form a right handed system and $\left| \overrightarrow{\mathbf{a}} \right| = 1,\ \left| \overrightarrow{\mathbf{b}} \right| = 3\ and\ \left| \overrightarrow{\mathbf{c}} \right| = 5,\ then\ \lbrack\ \overrightarrow{\mathbf{a}}\mathbf{-}2\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{b}}\mathbf{-}3\overrightarrow{\mathbf{c}}\mathbf{\ }\overrightarrow{\mathbf{c}}\mathbf{-}4\overrightarrow{\mathbf{a}}\ \rbrack$ is equal to
Explanation
(d)
Given that, $\left| \overrightarrow{\mathbf{a}} \right| = 1,\ \left| \overrightarrow{\mathbf{b}} \right| = 3\ and\ \left| \overrightarrow{\mathbf{c}} \right| = 5$
$\therefore\lbrack\ \overrightarrow{\mathbf{a}} - 2\overrightarrow{\mathbf{b}}\mathbf{\ }\overrightarrow{\mathbf{b}} - 3\overrightarrow{\mathbf{c}}\mathbf{\ }\overrightarrow{\mathbf{c}} - 4\overrightarrow{\mathbf{a}}\ \rbrack$
$= \left( \overrightarrow{\mathbf{a}} - 2\overrightarrow{\mathbf{b}} \right) \bullet \left\{ \ \left( \ \overrightarrow{\mathbf{b}} - 3\overrightarrow{\mathbf{c}}\ \right) \times \left( \overrightarrow{\mathbf{c}} - 4\overrightarrow{\mathbf{a}} \right) \right\}$
$= \left( \overrightarrow{\mathbf{a}} - 2\overrightarrow{\mathbf{b}} \right) \bullet \left\{ \ \overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{c}} - 4\overrightarrow{\mathbf{b}}\mathbf{\times}\overrightarrow{\mathbf{a}} + 12\overrightarrow{\mathbf{c}}\mathbf{\times}\overrightarrow{\mathbf{a}}\ \right\}$
$= \left( \overrightarrow{\mathbf{a}} - 2\overrightarrow{\mathbf{b}} \right) \bullet \left( \overrightarrow{\mathbf{a}} + 4\overrightarrow{\mathbf{c}} + 12\overrightarrow{\mathbf{b}}\ \right)\$
$= \overrightarrow{\mathbf{a}}\mathbf{\bullet}\overrightarrow{\mathbf{a}} - 24\overrightarrow{\mathbf{b}}\mathbf{\bullet}\overrightarrow{\mathbf{b}} = 1 - 24 \times 9$
$= 1 - 216 = - 215$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.