Answer
\[\left\langle -14,-14,14 \right\rangle \]
Work Step by Step
\[\begin{align}
& \text{Let }\mathbf{u}=\left\langle 6,-2,4 \right\rangle \text{ and }\mathbf{v}=\left\langle 1,2,3 \right\rangle \\
& \text{A vector orthogonal to }\mathbf{u}\text{ and }\mathbf{v}\text{ is parallel to }\mathbf{u}\times \mathbf{v}\text{. One such } \\
& \text{orthogonal vector is} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
6 & -2 & 4 \\
1 & 2 & 3 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
-2 & 4 \\
2 & 3 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
6 & 4 \\
1 & 3 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
6 & -2 \\
1 & 2 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=\left( -6-8 \right)\mathbf{i}-\left( 18-4 \right)\mathbf{j}+\left( 12+2 \right)\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=-14\mathbf{i}-14\mathbf{j}+14\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=\left\langle -14,-14,14 \right\rangle \\
\end{align}\]
