Scalar and vector product of two vectorsHard

Unit Vector and Vector Angles

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 α is :

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