Answer
$\frac{-\textbf{i}-4\textbf{j}+7\textbf{k}}{\sqrt {66}}$ and
$\frac{\textbf{i}+4\textbf{j}-7\textbf{k}}{\sqrt {66}}$
Work Step by Step
$\textbf{a}\times\textbf{b}$ and $\textbf{b}\times\textbf{a}$ are orthogonal to both $\textbf{a}$ and $\textbf{b}$.
The unit vector in the direction of $\textbf{a}\times\textbf{b}$ is obtained by dividing $\textbf{a}\times\textbf{b}$ by its magnitude.
The unit vector in the direction of $\textbf{b}\times\textbf{a}$ is obtained by dividing $\textbf{b}\times\textbf{a}$ by its magnitude.
Let the two unit vectors orthogonal to both $\textbf{a}$ and $\textbf{b}$ be $\textbf{u}$ and $\textbf{v}$. Then,
$\textbf{u}=\frac{\textbf{a}\times\textbf{b}}{||\textbf{a}\times\textbf{b}||}$
$\textbf{v}=\frac{\textbf{b}\times\textbf{a}}{||\textbf{b}\times\textbf{a}||}$
$\textbf{a}\times\textbf{b}=\begin{vmatrix}\textbf{i}&\textbf{j}&\textbf{k}\\3&1&1\\-1&2&1\end{vmatrix}$
$=\textbf{i}(1-2)-\textbf{j}(3+1)+\textbf{k}(6+1)$
$=-\textbf{i}-4\textbf{j}+7\textbf{k}$
$\textbf{b}\times\textbf{a}=\begin{vmatrix}\textbf{i}&\textbf{j}&\textbf{k}\\-1&2&1\\3&1&1\end{vmatrix}=$
$=\textbf{i}(2-1)-\textbf{j}(-1-3)+\textbf{k}(-1-6)$
$=\textbf{i}+4\textbf{j}-7\textbf{k}$
$||\textbf{a}\times\textbf{b}||=||\textbf{b}\times\textbf{a}||=\sqrt {1^{2}+4^{2}+7^{2}}=\sqrt {66}$
Therefore,
$\textbf{u}=\frac{-\textbf{i}-4\textbf{j}+7\textbf{k}}{\sqrt {66}}$ and
$\textbf{v}=\frac{\textbf{i}+4\textbf{j}-7\textbf{k}}{\sqrt {66}}$
