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Question
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4\hat{k} \text{ and } \vec{b} = 3 \hat{i} - 2 \hat{j} +\lambda \hat{k}\]
Solution
\[ \text{ If the vectors } \vec{a} \text{ and } \vec{b} \text{ are perpendicular to each other, then }\]
\[ \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \left( 2 \hat{i} + 3\hat{j} + 4 \hat{k} \right) . \left( 3 \hat{i} + 2 \hat{j} - \lambda \hat{k} \right) = 0\]
\[ \Rightarrow 6 + 6 - 4\lambda = 0\]
\[ \Rightarrow 12 - 4\lambda = 0\]
\[ \Rightarrow 4\lambda = 12\]
\[ \Rightarrow \lambda = 3\]
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