Let a = i +α j +β k, α, β ∈ R. Let a vector b be such that the angle between a and b is (π)/4 and | b |^2=6. If a · b =3 √ 2, then the value of…

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