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Question
A unit vector perpendicular to both \[\hat{ i } + \hat{ j } \text{ and } \hat{ j } + \hat{ k } \] is
Options
\[\hat{ i } - \hat{ j } + \hat{ k } \]
\[\hat{ i } + \hat{ j } + \hat{ k } \]
\[ \frac1 {\sqrt3} ( \hat{ i } + \hat{ j } + \hat{ k } ) \]
\[ \frac1 {\sqrt3} ( \hat{ i } - \hat{ j } + \hat{ k } ) \]
Solution
\[ \frac1 {\sqrt3} ( \hat{ i } - \hat{ j } + \hat{ k } ) \]
\[\text{ Let } :\]
\[ \vec{a} = \hat{ i } + \hat{ j } + 0 \hat{ k } \]
\[ \vec{b} = 0 \hat{ i } + \hat{ j } + \hat{ k } \]
\[ \therefore \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{vmatrix}\]
\[ = \hat{ i } - \hat{ j } + \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 1 + 1}\]
\[ = \sqrt{3}\]
\[\text{ Unit vector perpendicular to } \vec{a} \text{ and } \vec{b} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{\hat{ i } - \hat{ j } + \hat{ k } }{\sqrt{3}}\]
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