Vector Algebra: JEE Main 2021 Mathematics question

For p > 0, a vector ${\overrightarrow v _2} = 2\widehat i + (p + 1)\widehat j$ is obtained by rotating the vector ${\overrightarrow v _1} = \sqrt 3 p\widehat i + \widehat j$ by an angle $\theta$ about origin in counter clockwise direction. If $\tan \theta = {{\left( {\alpha \sqrt 3 - 2} \right)} \over {\left( {4\sqrt 3 + 3} \right)}}$, then the value of $\alpha$ is equal to _____________.