Let a = i +α j +3 k and b =3 i -α j + k. If the area of the parallelogram whose adjacent sides are represented by the vectors a and b is 8 √ 3 square…

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