Q. If the volume of a parallelopiped with $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}},\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}},\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$ as coterminous edges is 9 cu units, then the volume of the parallelopiped with $(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}) \times$($\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}),\ \left( \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}} \right) \times \left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right)$,$\left( \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right) \times (\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})$ as coterminous edges is
Explanation
(c)
Let $\overrightarrow{\mathbf{A}} = \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}},\ \overrightarrow{\mathbf{B}} = \overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}},\ \overrightarrow{\mathbf{C}} = \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}$
Given, $\left\lbrack \overrightarrow{\mathbf{A}}\ \overrightarrow{\mathbf{B}}\ \overrightarrow{\mathbf{C}} \right\rbrack = 9\$cu units
Using the relation $\left\lbrack \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}\ \overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{C}}\ \overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}} \right\rbrack = \left\lbrack \overrightarrow{\mathbf{A}}\ \overrightarrow{\mathbf{B}}\ \overrightarrow{\mathbf{C}} \right\rbrack^{2} = {(9)}^{2} = 81$ cu units
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.