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