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

Here, $\vec a \times \vec b = \left| {\begin{array}{llllllllllllllllllll}{\hat i}&{\hat j}&{\hat k}\\1&{ - 1}&3\\2&{ - 7}&1\end{array}} \right|$

$= ( - 1 + 21)\hat i - (i - 6)\hat j + ( - 7 + 2)\hat k = 20\hat i + 5\hat j - 5\hat k$

$\therefore$ Area of parallelogram $= |\vec a \times \vec b|$

$= |20\hat i + 5\hat j - 5\hat k| = \sqrt {{{(20)}^2} + {{(5)}^2} + {{( - 5)}^2}}$

$= \sqrt {400 + 25 + 25} = \sqrt {450} = 15\sqrt 2$ sq. units