Vector Algebra

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 _____________.