Let the vectors \(\vec{a}=-\hat{i}+\hat{j}+3 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}+\hat{k}\). For some \(\lambda ,\m…

Let the vectors \(\vec{a}=-\hat{i}+\hat{j}+3 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}+\hat{k}\). For some \(\lambda ,\mu \in R\), let \(\vec{c}=\lambda \vec{a}+\mu \vec{b}\). If \(\vec{c} \cdot(3 \hat{i}-6 \hat{j}+2 \hat{k})=10\) and \(\vec{c} \cdot(\hat{i}+\hat{j}+\hat{k})=-2\), then \(|\vec{c}|^2\) is equal to:

[JEE Main 2026, 2 Apr (Shift 2)]