Let \(\vec{a}=\hat{i}+\alpha \hat{j}+\beta \hat{k}, \alpha, \beta \in R\). Let a vector \(\vec{b}\) be such that the ang…

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

[JEE Main 2024, 30 Jan (Shift 2)]