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Question
For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if
\[\vec{a} = \lambda \hat{i} + 2\hat{j} + \hat{k} \text{ and } \vec{b} = 5\hat{i} - 9 \hat{j} + 2\hat{k}\]
Solution
\[\text{ If the vectors } \vec{a} \text{ and } \vec{b} \text{ are perpendicular to each other, then }\]
\[ \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \left( \lambda \hat{i} + 2 \hat{j} + \hat{k} \right) . \left( 5 \hat{i} - 9 \hat{j} + 2 \hat{k}\right) = 0\]
\[ \Rightarrow 5\lambda - 18 + 2 = 0\]
\[ \Rightarrow 5\lambda - 16 = 0\]
\[ \Rightarrow 5\lambda = 16\]
\[ \Rightarrow \lambda = \frac{16}{5}\]
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