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 \h…

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 \(\cos ^{-1}\left(-\frac{1}{3}\right)\) with the vector \(\hat{i}+\hat{j}+\hat{k}\) . If \(\hat{a}\) makes an angle of \(\frac{\pi}{3}\) with the vector \(\hat{i}+\alpha \hat{j}+\hat{k}\) , then the value of \(\alpha\) is :