Answer
\[\frac{8}{\sqrt{377}}\mathbf{i}+\frac{12}{\sqrt{377}}\mathbf{j}+\frac{13}{\sqrt{377}}\mathbf{k}\]
Work Step by Step
\[\begin{align}
& \text{Let }\mathbf{u}=\left\langle 2,-10,8 \right\rangle \text{ and }\mathbf{v}=\left\langle 4,6,-8 \right\rangle \\
& \text{Find }\mathbf{u}\times \mathbf{v} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
2 & -10 & 8 \\
4 & 6 & -8 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
-10 & 8 \\
6 & -8 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
2 & 8 \\
4 & -8 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
2 & -10 \\
4 & 6 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=32\mathbf{i}+48\mathbf{j}+52\mathbf{k} \\
& \\
& \text{Finding a unit vector that is orthogonal to both }\mathbf{u}\text{ and }\mathbf{v} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{32\mathbf{i}+48\mathbf{j}+52\mathbf{k}}{\sqrt{{{\left( 32 \right)}^{2}}+{{\left( 48 \right)}^{2}}+{{\left( 52 \right)}^{2}}}} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{1}{\sqrt{6032}}\left( 32\mathbf{i}+48\mathbf{j}+52\mathbf{k} \right) \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{1}{4\sqrt{377}}\left( 32\mathbf{i}+48\mathbf{j}+52\mathbf{k} \right) \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{32}{4\sqrt{377}}\mathbf{i}+\frac{48}{4\sqrt{377}}\mathbf{j}+\frac{52}{4\sqrt{377}}\mathbf{k} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{8}{\sqrt{377}}\mathbf{i}+\frac{12}{\sqrt{377}}\mathbf{j}+\frac{13}{\sqrt{377}}\mathbf{k} \\
\end{align}\]
