Answer
\[\begin{align}
& \mathbf{u}\times \mathbf{v}=8\mathbf{i}+4\mathbf{j}+10\mathbf{k} \\
& \mathbf{v}\times \mathbf{u}=-8\mathbf{i}-4\mathbf{j}-10\mathbf{k} \\
\end{align}\]
Work Step by Step
\[\begin{align}
& \mathbf{u}=3\mathbf{i}-\mathbf{j}-2\mathbf{k},\text{ }\mathbf{v}=\mathbf{i}+3\mathbf{j}-2\mathbf{k} \\
& \text{Calculate the cross product }\mathbf{u}\times \mathbf{v} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
3 & -1 & -2 \\
1 & 3 & -2 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
-1 & -2 \\
3 & -2 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
3 & -2 \\
1 & -2 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
3 & -1 \\
1 & 3 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=\left( 2+6 \right)\mathbf{i}-\left( -6+2 \right)\mathbf{j}+\left( 9+1 \right)\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=8\mathbf{i}+4\mathbf{j}+10\mathbf{k} \\
& \text{Calculate the cross product }\mathbf{v}\times \mathbf{u} \\
& \mathbf{v}\times \mathbf{u}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & 3 & -2 \\
3 & -1 & -2 \\
\end{matrix} \right| \\
& \mathbf{v}\times \mathbf{u}=\left| \begin{matrix}
3 & -2 \\
-1 & -2 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
1 & -2 \\
3 & -2 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
1 & 3 \\
3 & -1 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{v}\times \mathbf{u}=\left( -6-2 \right)\mathbf{i}-\left( -2+6 \right)\mathbf{j}+\left( -1-9 \right)\mathbf{k} \\
& \mathbf{v}\times \mathbf{u}=-8\mathbf{i}-4\mathbf{j}-10\mathbf{k} \\
\end{align}\]
