Let \(\overrightarrow a = 2\hat i + \hat j - \hat k \) and \(\overrightarrow b = \hat i + 2\hat j + \hat k \) be two vectors. Consider a vector \(\overrightarrow c = \alpha \overrightarrow a + \beta \overrightarrow b ,\alpha ,\beta \in R. \) If the projection of \(\overrightarrow{c}\) on the vector \((\vec{a}+\vec{b}) \) is \(3\sqrt{2} \), then the minimum value \((\vec{c}-(\vec{a}\times\vec{b})).\vec{c} \)  equals