Question

If \[\vec{a} = 2 \hat{ i } + \hat{ k }  , \vec{b} = \hat { i }  + \hat{ j } + \hat{ k }  ,\]  find the magnitude of  \[\vec{a} \times \vec{b} .\]

 

 

Answer 1

\[\text{ Given } : \]
\[ \vec{a} = 2 \hat{ i }  + 0 \hat{ j }  + \hat{ k }  \]
\[ \vec{b} = \hat{ i } + \hat{ j }  +\hat{  k } \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k }  \\ 2 & 0 & 1 \\ 1 & 1 & 1\end{vmatrix}\]
\[ = \left( 0 - 1 \right) \hat{ i } - \left( 2 - 1 \right) \hat{ j }  + \left( 2 - 0 \right) \hat{ k } \]
\[ = - \hat{ i }  - \hat{ j } + 2 \hat{ k }  \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{\left( - 1 \right)^2 + \left( - 1 \right)^2 + 2^2}\]
\[ = \sqrt{6}\]