Three honey bees were found flying along the vectors $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 4\hat{j} - 2\hat{k}$ and $\vec{c} = 3\hat{i} + 2\hat{k}$ respectively. Find the scalar value of $\lambda$ such that the path represented by the vector $\vec{a} + \lambda \vec{b}$ is perfectly perpendicular to the vector $\vec{c}$.
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When computing dot products where one component is missing (like $\hat{j}$ in vector $\vec{c}$), that entire middle term becomes zero automatically. Pay close attention to missing components to avoid calculation errors!