Explanation:
\(\vec A \cdot \overrightarrow B = \left| {\overrightarrow A } \right|\left| {\overrightarrow B } \right|\cos \theta = 2\sqrt 3 \,\,\,\,\,\,\,\,\,\,\,\,(1)\)
\(\left| {\overrightarrow A } \right| \times \left| {\overrightarrow B } \right| = \left| {\overrightarrow A } \right|\left| {\overrightarrow B } \right|\sin \theta = 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
Dividing equation (2) by equation (1) we get,
\(\tan \theta = \frac{2}{{2\sqrt 3 }} = \frac{1}{{\sqrt 3 }}\)
\(\therefore \theta = {\tan ^{ - 1}}\left( {\frac{1}{{\sqrt 3 }}} \right) = 30^\circ \)