Answer
\[6\sqrt {41} \]
Work Step by Step
\[\begin{gathered}
{\mathbf{u}} = 8{\mathbf{i}} + 2{\mathbf{j}} - 3{\mathbf{k}},\,\,\,{\mathbf{v}} = 2{\mathbf{i}} + 4{\mathbf{j}} - 4{\mathbf{k}} \hfill \\
{\text{The area of the parallelogram is given by }}\left| {{\mathbf{u}} \times {\mathbf{v}}} \right| \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
8&2&{ - 3} \\
2&4&{ - 4}
\end{array}} \right| \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left| {\begin{array}{*{20}{c}}
2&{ - 3} \\
4&{ - 4}
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
8&{ - 3} \\
2&{ - 4}
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
8&2 \\
2&4
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left( { - 8 + 12} \right){\mathbf{i}} - \left( { - 32 + 6} \right){\mathbf{j}} + \left( {32 - 4} \right){\mathbf{k}} \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = 4{\mathbf{i}} + 26{\mathbf{j}} + 28{\mathbf{k}} \hfill \\
{\text{Area}} = \left| {{\mathbf{u}} \times {\mathbf{v}}} \right| \hfill \\
{\text{Area}} = \left| {4{\mathbf{i}} + 26{\mathbf{j}} + 28{\mathbf{k}}} \right| \hfill \\
{\text{Area}} = \sqrt {{{\left( 4 \right)}^2} + {{\left( {26} \right)}^2} + {{\left( {28} \right)}^2}} \hfill \\
{\text{Area}} = \sqrt {1476} \hfill \\
{\text{Area}} = 6\sqrt {41} \hfill \\
\end{gathered} \]