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 \( \lvert \vec{d} - \vec{a} \rvert = \sqrt{11} \), \( \lvert \vec{c} \times \vec{d} \rvert = 3 \) and the angle between \( \vec{c} \) and \( \vec{d} \) is \( \frac{\pi}{4} \). Then \( \vec{a} \cdot \vec{d} \) is equal to:
