Answer
\[{\mathbf{u}} \times {\mathbf{v}} = \left\langle { - 8,9,10} \right\rangle {\text{ and }}{\mathbf{u}} \times {\mathbf{v}} = \left\langle {8, - 9, - 10} \right\rangle \]
Work Step by Step
\[\begin{gathered}
{\mathbf{u}} = \left\langle {3, - 4,6} \right\rangle ,\,\,\,{\mathbf{v}} = \left\langle {1,2, - 1} \right\rangle \hfill \\
\hfill \\
{\text{Calculate }}{\mathbf{u}} \times {\mathbf{v}} \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
3&{ - 4}&6 \\
1&2&{ - 1}
\end{array}} \right| \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left| {\begin{array}{*{20}{c}}
{ - 4}&6 \\
2&{ - 1}
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
3&6 \\
1&{ - 1}
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
3&{ - 4} \\
1&2
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left( {4 - 12} \right){\mathbf{i}} - \left( { - 3 - 6} \right){\mathbf{j}} + \left( {6 + 4} \right){\mathbf{k}} \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = - 8{\mathbf{i}} + 9{\mathbf{j}} + 10{\mathbf{k}} \hfill \\
{\mathbf{u}} \times {\mathbf{v}} = \left\langle { - 8,9,10} \right\rangle \hfill \\
and \hfill \\
{\mathbf{v}} \times {\mathbf{u}} = - {\mathbf{u}} \times {\mathbf{v}} \hfill \\
{\mathbf{v}} \times {\mathbf{u}} = - \left\langle { - 8,9,10} \right\rangle \hfill \\
{\mathbf{v}} \times {\mathbf{u}} = \left\langle {8, - 9, - 10} \right\rangle \hfill \\
\end{gathered} \]
