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