Let a be a unit vector perpendicular to the vectors b = i -2 j +3 k and c =2 i +3 j - k, and makes an angle of cos^-1(-1/3) with the vector i + j +…

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