Question

Write a unit vector perpendicular to \[\hat{ i } + \hat{ j }  \text{ and }  \hat{ j }  + \hat{ k } .\]

 

Answer 1

\[\text{ Let } \vec{a} = \hat{ i }  + \hat{ j }  + 0 \hat{ k }  ; \vec{b} = 0 \hat{ i  } + \hat{ j } + \hat{ k } \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }  & \hat{ j } & \hat{ k }  \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{vmatrix}\]
\[ = \hat{ i }  - \hat{ j }  + \hat{ k }  \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{1 + 1 + 1}\]
\[ = \sqrt{3}\]
\[\text{ Unit vector perpendicular to }  \vec{a} \text{ and } \vec{b} \text{ is } ,\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} = \frac{1}{\sqrt{3}}\left( \hat{ i }  - \hat{ j }  + \hat{ k }  \right)\]