Q. The unit vector perpendicular to $\widehat{\mathbf{i}}\mathbf{-}\widehat{\mathbf{j}}$ and coplanar with $\widehat{\mathbf{i}} + 2\widehat{\mathbf{j}}\ and\ 2\widehat{\mathbf{i}} + 3\widehat{\mathbf{j}}$ is
A) $\frac{2\widehat{\mathbf{i}} - 5\widehat{\mathbf{j}}}{\sqrt{2a}}$
B) $2\widehat{\mathbf{i}} + 5\widehat{\mathbf{j}}$
C) $\frac{1}{\sqrt{2}}(\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}})$
D) $\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}$
Explanation
(c)
Let the unit vector $\frac{\widehat{\mathbf{i}}\mathbf{-}\widehat{\mathbf{j}}}{\sqrt{2}}$ is perpendicular to $\widehat{\mathbf{i}}\mathbf{-}\widehat{\mathbf{j}}$, then we get
$\frac{\left( \widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}} \right)\mathbf{\bullet (}\widehat{\mathbf{i}}\mathbf{-}\widehat{\mathbf{j}})}{\sqrt{2}} = \frac{1 - 1}{\sqrt{2}} = 0$
$\therefore\frac{\widehat{\mathbf{i}}\mathbf{+}\widehat{\mathbf{j}}}{\sqrt{2}}\$ is the required unit vector.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.