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Question
Write the value of λ so that the vectors \[\vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}\] are perpendicular to each other.
Solution
\[\text{ We have }\]
\[ \vec{a} = 2 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k} \]
\[\text{ The given vectors are perpendicular. So, their dot product is zero. }\]
\[\left( 2 \hat{i} + \lambda \hat{j} + \hat{k} \right) . \left( \hat{i} - 2 \hat{j} + 3 \hat{k} \right)\]
\[ \Rightarrow 2 - 2\lambda + 3 = 0\]
\[ \Rightarrow 5 - 2\lambda = 0\]
\[ \Rightarrow - 2\lambda = - 5\]
\[ \Rightarrow \lambda = \frac{5}{2}\]
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