Answer
\[\left\langle 3,-4,2 \right\rangle \]
Work Step by Step
\[\begin{align}
& \text{Let }\mathbf{u}=\left\langle 0,1,2 \right\rangle \text{ and }\mathbf{v}=\left\langle -2,0,3 \right\rangle \\
& \text{A vector orthogonal to }\mathbf{u}\text{ and }\mathbf{v}\text{ is parallel to }\mathbf{u}\times \mathbf{v}\text{. One such } \\
& \text{orthogonal vector is} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
0 & 1 & 2 \\
-2 & 0 & 3 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
1 & 2 \\
0 & 3 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
0 & 2 \\
-2 & 3 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
0 & 1 \\
-2 & 0 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=\left( 3-0 \right)\mathbf{i}-\left( 0+4 \right)\mathbf{j}+\left( 0+2 \right)\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=3\mathbf{i}-4\mathbf{j}+2\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=\left\langle 3,-4,2 \right\rangle \\
\end{align}\]
