Let a =α i +2 j - k and b =-2 i +α j + k, where α ∈ R. If the area of the parallelogram whose adjacent sides are represented by the vectors a and b…

Let $\displaystyle \overrightarrow{\mathrm{a}}=\alpha \hat{i}+2 \hat{j}-\hat{k}$ and $\displaystyle \overrightarrow{\mathrm{b}}=-2 \hat{i}+\alpha \hat{j}+\hat{k}$, where $\displaystyle \alpha \in \mathbf{R}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$ is $\displaystyle \sqrt{15\left(\alpha^2+4\right)}$, then the value of $\displaystyle 2|\overrightarrow{\mathrm{a}}|^2+(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}})|\overrightarrow{\mathrm{b}}|^2$ is equal to :