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