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Question
Find the value of λ if the vectors \[2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \text{ and } 3 \hat{i} + 2 \hat{j} - 4 \hat{k}\] are perpendicular to each other.
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Solution
\[\text{ Given }: 2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \text{ and } 3i + 2j - 4k \text{ are perpendicular to each other } . \]
\[\text{ So, their dot product is zero }.\]
\[\left( 2 \hat{i} + \lambda \hat{j} + 3 \hat{k} \right) . \left( 3i + 2j - 4k \right)\]
\[ \Rightarrow 6 + 2\lambda - 12 = 0\]
\[ \Rightarrow 2\lambda - 6 = 0\]
\[ \Rightarrow \lambda = 3\]
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