Question

Find λ, if \[\left( 2 \hat{ i }  + 6 \hat{ j }  + 14 \hat{ k }  \right) \times \left( \hat{ i }  - \lambda \hat{ j } + 7 \hat{ k }  \right) = \vec{0} .\]

 

Answer 1

\[\text{ Given } \begin{vmatrix} \hat{ i}  & \hat{ j }  & \hat{ k }  \\ 2 & 6 & 14 \\ 1 & - \lambda & 7\end{vmatrix} = \vec{0} \]
\[ \Rightarrow \hat{ i }  \left( 42 + 14\lambda \right) - 0 \hat{ j }  + \hat{ k }  \left( - 2\lambda - 6 \right) = 0 \hat{ i } + 0 \hat{ j }  + 0 \hat{ k }  \]
\[ \Rightarrow 42 + 14\lambda = 0; - 2\lambda - 6 = 0\]
\[ \Rightarrow \lambda = - 3 (\text{ This satisfies the above equations } ) \]