Q. If $\overrightarrow{\mathbf{a}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}2\widehat{\mathbf{j}}\mathbf{+}3\widehat{\mathbf{k}}\ and\ \overrightarrow{\mathbf{b}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{\times}\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\widehat{\mathbf{i}} \right)\mathbf{+}\widehat{\mathbf{j}}\mathbf{\times}\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\widehat{\mathbf{j}} \right)\mathbf{+}\widehat{\mathbf{k}}\mathbf{\times (}\overrightarrow{\mathbf{a}}\mathbf{\times}\widehat{\mathbf{k}})$, then length of $\overrightarrow{\mathbf{b}}$ is equal to
Explanation
(d)
We have, $\overrightarrow{\mathbf{a}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{+}2\widehat{\mathbf{j}}\mathbf{+}3\widehat{\mathbf{k}}$
and $\overrightarrow{\mathbf{b}}\mathbf{=}\widehat{\mathbf{i}}\mathbf{\times}\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\widehat{\mathbf{i}} \right)\mathbf{+}\widehat{\mathbf{j}}\mathbf{\times}\left( \overrightarrow{\mathbf{a}}\mathbf{\times}\widehat{\mathbf{j}} \right)\mathbf{+}\widehat{\mathbf{k}}\mathbf{\times (}\overrightarrow{\mathbf{a}}\mathbf{\times}\widehat{\mathbf{k}})$
$= 3\overrightarrow{\mathbf{a}}\mathbf{-}\overrightarrow{\mathbf{a}} = 2\overrightarrow{\mathbf{a}}$
$= 2(\widehat{\mathbf{i}}\mathbf{+}2\widehat{\mathbf{j}}\mathbf{+}3\widehat{\mathbf{k}})$
$\Rightarrow \left| \overrightarrow{\mathbf{b}} \right| = \sqrt{4 + 16 + 36} = \sqrt{56} = 2\sqrt{14}\$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.