Question

Find a vector of magnitude 49, which is perpendicular to both the vectors  \[2 \hat{ i }   + 3 \hat{ j }  + 6 \hat{ k }  \text{ and } 3 \hat{ i }  - 6 \hat{ j }  + 2 \hat{ k }  .\]

 

Answer 1

\[\text{ Given } : \]
\[ \vec{a} = 2 \hat{ i }  + 3 \hat{ j }  + 6 \hat{ k } \]
\[ \vec{b} = 3 \hat{ i }  - 6 \hat{ j }  + 2 \hat{ k }  \]
\[ \vec{a} \times \vec{b} = \begin{vmatrix}\hat{ i }  & \hat{ j }  & \hat{ k  } \\ 2 & 3 & 6 \\ 3 & - 6 & 2\end{vmatrix}\] 
\[ = \left( 6 + 36 \right) \hat{ i  } - \left( 4 - 18 \right) \hat{ j }  + \left( - 12 - 9 \right) \hat{ k }  \]
\[ = 42 \hat{ i }  + 14 \hat{ j }  - 21 \hat{ k } \]
\[ \Rightarrow \left| \vec{a} \times \vec{b} \right| = \sqrt{{42}^2 + {14}^2 + \left( - {21}^2 \right)}\]
\[ = \sqrt{2401}\]
\[ = 49\]
\[\text{ Required vector } = 49 \times \left\{ \frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|} \right\}\]
\[ = 49 \times \frac{42 \hat{ i }  + 14 \hat{ j }  - 21 \hat{ k } }{49}\]
\[ = 42 \hat{ i }  + 14 \hat{ j }  - 21 \hat{ k }  \]