Let \( \vec{a} = \hat{\imath} + \alpha \hat{\jmath} + \beta \hat{k}, \ \alpha,\beta \in \mathbb{R} \). Let a vector \( \vec{b} \) be such that the angle between \( \vec{a} \) and \( \vec{b} \) is \( \dfrac{\pi}{4} \) and \(|\vec{b}|^{2} = 6 \). If \( \vec{a} \cdot \vec{b} = 3\sqrt{2} \), then the value of \( (\alpha^{2} + \beta^{2}) \, |\vec{a} \times \vec{b}|^{2} \) is equal to:
