Find $|\vec a \times \vec b|$, if $\vec a = \hat i - 7\hat j + 7\hat k$ and $\vec b = 3\hat i - 2\hat j + 2\hat k$.

We have, $\vec a = \hat i - 7\hat j + 7\hat k$ and $\vec b = 3\hat i - 2\hat j + 2\hat k$

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

$= ( - 14 + 14)\hat i - (2 - 21)\hat j + ( - 2 + 21)\hat k = 19\hat j + 19\hat k$

$\therefore$ $|\vec a \times \vec b| = \sqrt {{{(19)}^2} + {{(19)}^2}} = 19\sqrt 2$