Let $\overrightarrow { \mathrm { a } } = \hat { \mathrm { i } } + \alpha \hat { \mathrm { j } } + 3 \hat { \mathrm { k } }$ and $\overrightarrow { \mathrm { b } } = 3 \hat { \mathrm { i } } - \alpha \hat { \mathrm { j } } + \hat { \mathrm { k } }$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec { a }$ and $\vec { b }$ is $8 \sqrt { 3 }$ square units, then $\vec { a } \cdot \vec { b }$ is equal to $\underline{\hspace{1cm}}$.