Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat{i}-\hat{j}+3\hat{k}$ and $\vec{b}=2\hat{i}-7\hat{j}+\hat{k}$.

Let $\bar{a}, \bar{b}, \bar{c}$ be vectors such that $\bar{a} \neq \bar{o}, \bar{b} \neq \bar{o}, \bar{a} \times \bar{c} = \bar{b}$ and $\bar{b} \times \bar{c} = \bar{a}$. Then:

If $\overrightarrow{a}+2 \overrightarrow{b}+3 \overrightarrow{c}=\overrightarrow{0}$,then $\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{c} \times \overrightarrow{a}$ is equal to

If $a = 2i + 3j - 5k$,$b = mi + nj + 12k$ and $a \times b = 0$,then $(m, n) = $

If $\overrightarrow{OA} = 3i + 2j - k$ and $\overrightarrow{OB} = i + 3j + k$,then the area of the triangle $OAB$ is

The unit vector perpendicular to the vector $\hat{i}-2 \hat{j}+3 \hat{k}$ and coplanar with the vectors $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}-\hat{j}-\hat{k}$ is