Answer
\[\left\langle -\frac{7}{9\sqrt{3}},-\frac{5}{9\sqrt{3}},\frac{13}{9\sqrt{3}} \right\rangle \]
Work Step by Step
\[\begin{align}
& \text{Let the vectors be: }\mathbf{u}=\left\langle 4,-3,1 \right\rangle ,\text{ }\mathbf{v}=\left\langle 2,5,3 \right\rangle \\
& \\
& \text{Find }\mathbf{u}\times \mathbf{v} \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & -3 & 1 \\
2 & 5 & 3 \\
\end{matrix} \right| \\
& \mathbf{u}\times \mathbf{v}=\left| \begin{matrix}
-3 & 1 \\
5 & 3 \\
\end{matrix} \right|\mathbf{i}-\left| \begin{matrix}
4 & 1 \\
2 & 3 \\
\end{matrix} \right|\mathbf{j}+\left| \begin{matrix}
4 & -3 \\
2 & 5 \\
\end{matrix} \right|\mathbf{k} \\
& \mathbf{u}\times \mathbf{v}=-14\mathbf{i}-10\mathbf{j}+26\mathbf{k} \\
& \\
& \text{Finding a unit vector that is orthogonal to both }\mathbf{u}\text{ and }\mathbf{v} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{-14\mathbf{i}-10\mathbf{j}+26\mathbf{k}}{\sqrt{{{\left( -14 \right)}^{2}}+{{\left( -10 \right)}^{2}}+{{\left( 26 \right)}^{2}}}} \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\frac{1}{\sqrt{972}}\left\langle -14,-10,26 \right\rangle \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\left\langle -\frac{14}{\sqrt{972}},-\frac{10}{\sqrt{972}},\frac{26}{\sqrt{972}} \right\rangle \\
& \frac{\mathbf{u}\times \mathbf{v}}{\left\| \mathbf{u}\times \mathbf{v} \right\|}=\left\langle -\frac{7}{9\sqrt{3}},-\frac{5}{9\sqrt{3}},\frac{13}{9\sqrt{3}} \right\rangle \\
\end{align}\]
