Let a =2 i +α j + k, b =- i + k, c =β j - k, where α and β are integers and α β=-6. Let the values of the ordered pair ( α, β ), for which the area…

Let $\displaystyle \vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}$, where $\displaystyle \alpha$ and $\displaystyle \beta$ are integers and $\displaystyle \alpha \beta=-6$. Let the values of the ordered pair ( $\displaystyle \alpha, \beta$ ), for which the area of the parallelogram of diagonals $\displaystyle \vec{a}+\vec{b}$ and $\displaystyle \vec{b}+\vec{c}$ is $\displaystyle \frac{\sqrt{21}}{2}$, be $\displaystyle \left(\alpha_1, \beta_1\right)$ and $\displaystyle \left(\alpha_2, \beta_2\right)$. Then $\displaystyle \alpha_1^2+\beta_1^2-\alpha_2 \beta_2$ is equal to