Let \( \vec{a} = 2\hat{i} + \alpha \hat{j} + \hat{k}, \ \vec{b} = -\hat{i} + \hat{k}, \ \vec{c} = \beta \hat{j} - \hat{k} \), where \( \alpha \) and \( \beta \) are integers and \( \alpha \beta = -6 \). Let the values of the ordered pair \( (\alpha , \beta) \), for which the area of the parallelogram of diagonals \( \vec{a} + \vec{b} \) and \( \vec{b} + \vec{c} \) is \( \frac{\sqrt{21}}{2} \), be \( (\alpha_{1}, \beta_{1}) \) and \( (\alpha_{2}, \beta_{2}) \). Then \( \alpha_{1}^{2} + \beta_{1}^{2} - \alpha_{2}\beta_{2} \) is equal to
