For \( p>0 \), a vector \( \vec{v}_{2}=2\hat{i}+(p+1)\hat{j} \) is obtained by rotating the vector \( \vec{v}_{1}=\sqrt{3}\,p\,\hat{i}+\hat{j} \) by an angle \( \theta \) about origin in counter clockwise direction. If \( \tan\theta=\dfrac{(\alpha\sqrt{3}-2)}{(4\sqrt{3}+3)} ,\) then the value of \( \alpha \) is equal to ______.
