Let \( \hat{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \(\displaystyle
\cos^{-1}\!\left(-\frac{1}{3}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \hat{a} \) makes an angle of \(\displaystyle
\frac{\pi}{3} \) with the vector \( \hat{i} + \alpha \hat{j} + \hat{k} \), then the value of \( \alpha \) is:
