Q. Let $\overrightarrow{\mathbf{u}} = \widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}},\ \overrightarrow{\mathbf{v}} = \ \widehat{\dot{\mathbf{i}}} - \widehat{\dot{\mathbf{j}}}$ and $\overrightarrow{\mathbf{w}} = \widehat{\dot{\mathbf{i}}} + 2\widehat{\dot{\mathbf{j}}} + 3\widehat{\mathbf{k}}$. If $\widehat{\mathbf{n}}$ is a unit vector such that $\overrightarrow{\mathbf{u}}.\widehat{\mathbf{n}} = 0$ and $\overrightarrow{\mathbf{v}}.\widehat{\mathbf{n}} = 0$, then $\left| \overrightarrow{w}.\widehat{\mathbf{n}} \right|$ is equal to
Explanation
(d)
$\overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}} = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
1 & 1 & 0 \\
1 & - 1 & 0
\end{matrix} \right| = - 2\widehat{\mathbf{k}}$
$\therefore\left| \overrightarrow{\mathbf{w}} \bullet \widehat{\mathbf{n}} \right| = \frac{\left| \overrightarrow{\mathbf{w}} \bullet \overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}} \right|}{\left| \overrightarrow{\mathbf{u}} \times \overrightarrow{\mathbf{v}} \right|}$
$\Longrightarrow \left| \overrightarrow{\mathbf{w}} \bullet \widehat{\mathbf{n}} \right| = \frac{| - 6\widehat{\mathbf{k}}|}{| - 2\widehat{\mathbf{k}}|} = 3$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.