Let \(\vec{a}=2\hat{i}+\hat{j}−2\hat{k},\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\vec{a}\times \vec{b}.\) Let \(\vec{d}\)…

Let \(\vec{a}=2\hat{i}+\hat{j}−2\hat{k},\vec{b}=\hat{i}+\hat{j}\) and \(\vec{c}=\vec{a}\times \vec{b}.\) Let \(\vec{d}\) be a vector such that \(|\vec{d}-\vec{a}|=\sqrt{11},|\vec{c} \times \vec{d}|=3\) and the angle between \(\vec{c}\) and \(\vec{d}\) is \(\frac{\pi }{4}.\) Then \(\vec{a}⋅\vec{d}\) is equal to