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

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\overline{\mathrm{c}}=\hat{\mathrm{a}}+2 \hat{\mathrm{~b}}$ and $\overline{\mathrm{d}}=5 \hat{\mathrm{a}}+4 \hat{\mathrm{~b}}$ are perpendicular to each other, then the angle between $\hat{a}$ and $\hat{b}$ is

If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}, \hat{i}+\hat{j}+c \hat{k}$ $(a \neq b, c \neq 1)$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ has the value __________.

If $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ are three non-coplanar vectors, then $(\bar{a}+\bar{b}+\bar{c}) \cdot[(\bar{a}+\bar{b}) \times(\bar{a}+\bar{c})]$ equals