Question

For what value of λ are the vectors \[\vec{a} \text{ and } \vec{b}\] perpendicular to each other if

\[\vec{a} = 2 \hat{i} + 3 \hat{j} + 4\hat{k} \text{ and } \vec{b} = 3 \hat{i} - 2 \hat{j} +\lambda \hat{k}\]

Answer 1

\[ \text{ If the vectors } \vec{a} \text{ and } \vec{b} \text{ are perpendicular to each other, then }\]

\[ \vec{a} . \vec{b} = 0\]

\[ \Rightarrow \left( 2 \hat{i} + 3\hat{j} + 4 \hat{k} \right) . \left( 3 \hat{i} + 2 \hat{j} - \lambda \hat{k} \right) = 0\]

\[ \Rightarrow 6 + 6 - 4\lambda = 0\]

\[ \Rightarrow 12 - 4\lambda = 0\]

\[ \Rightarrow 4\lambda = 12\]

\[ \Rightarrow \lambda = 3\]