Answer
\[\left\langle \pm \frac{3}{\sqrt{59}},\pm \frac{7}{\sqrt{59}},\pm \frac{1}{\sqrt{59}} \right\rangle \]
Work Step by Step
\[\begin{align}
& \text{Let the vectors: }\mathbf{u}=-3\mathbf{i}+2\mathbf{j}-5\mathbf{k},\text{ }\mathbf{v}=\mathbf{i}-\mathbf{j}+4\mathbf{k} \\
& \\
& \text{Find }\mathbf{u}\times \mathbf{v} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-3 & 2 & -5 \\
1 & -1 & 4 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
2 & -5 \\
-1 & 4 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
-3 & -5 \\
1 & 4 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
-3 & 2 \\
1 & -1 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=3\mathbf{i}+7\mathbf{j}+\mathbf{k} \\
& \\
& \text{Finding a unit vector that is orthogonal to both }\mathbf{u}\text{ and }\mathbf{v} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\pm \frac{3\mathbf{i}+7\mathbf{j}+\mathbf{k}}{\sqrt{{{\left( 3 \right)}^{2}}+{{\left( 7 \right)}^{2}}+{{\left( 1 \right)}^{2}}}} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\pm \frac{1}{\sqrt{59}}\left( 3\mathbf{i}+7\mathbf{j}+\mathbf{k} \right) \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\pm \frac{3}{\sqrt{59}}\mathbf{i}+\frac{7}{\sqrt{59}}\mathbf{j}+\frac{1}{\sqrt{59}}\mathbf{k} \\
& or \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\left\langle \pm \frac{3}{\sqrt{59}},\pm \frac{7}{\sqrt{59}},\pm \frac{1}{\sqrt{59}} \right\rangle \\
\end{align}\]
