Vector Algebra

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 $$\left(\alpha_1, \beta_1\right)$$ and $$\left(\alpha_2, \beta_2\right)$$. Then $$\alpha_1^2+\beta_1^2-\alpha_2 \beta_2$$ is equal to