Mathematics - Vector Question with Solution | TestHub

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