Let \(\vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\vec{b}=-\hat{i}+2\hat{j}+\hat{k}\) and \(\vec{c}=3\hat{i}+\hat{j}\). Then the value of \(\lambda\) for which \(\vec{a}+\lambda \vec{b}\) is perpendicular to \(\vec{c}\) is

\(6\) \(8\) \(10\) \(12\)

\(8\)

\(\begin{aligned}\left(\vec{a}+\lambda \vec{b}\right)\cdot \vec{c}&=0\\ 3(2-\lambda)+(2+2\lambda)&=0\\ 6-3\lambda +2+2\lambda &=0\\ \lambda &=8\end{aligned}\)