Answer
\[\frac{5}{3\sqrt{22}}\mathbf{i}+\frac{2}{3\sqrt{22}}\mathbf{j}+\frac{13}{3\sqrt{22}}\mathbf{k}\]
Work Step by Step
\[\begin{align}
& \text{Let the vectors be: }\mathbf{u}=\left\langle -8,-6,4 \right\rangle ,\text{ }\mathbf{v}=\left\langle 10,-12,-2 \right\rangle \\
& \\
& \text{Find }\mathbf{u}\times \mathbf{v} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-8 & -6 & 4 \\
10 & -12 & -2 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
-6 & 4 \\
-12 & -2 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
-8 & 4 \\
10 & -2 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
-8 & -6 \\
10 & -12 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=60\mathbf{i}+24\mathbf{j}+156\mathbf{k} \\
& \\
& \text{Finding a unit vector that is orthogonal to both }\mathbf{u}\text{ and }\mathbf{v} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{60\mathbf{i}+24\mathbf{j}+156\mathbf{k}}{\sqrt{{{\left( 60 \right)}^{2}}+{{\left( 24 \right)}^{2}}+{{\left( 156 \right)}^{2}}}} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{1}{\sqrt{28512}}\left( 60\mathbf{i}+24\mathbf{j}+156\mathbf{k} \right) \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{1}{36\sqrt{22}}\left( 60\mathbf{i}+24\mathbf{j}+156\mathbf{k} \right) \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{5}{3\sqrt{22}}\mathbf{i}+\frac{2}{3\sqrt{22}}\mathbf{j}+\frac{13}{3\sqrt{22}}\mathbf{k} \\
\end{align}\]
