Answer
$⟨10,-8,-7⟩$ and $⟨-10,8,7⟩$
Work Step by Step
Vectors orthogonal to both $\textbf{v}$ and $\textbf{w}$ are $\textbf{v}\times\textbf{w}$ and $\textbf{w}\times\textbf{v}$.
$\textbf{v}\times\textbf{w}=\begin{vmatrix}\textbf{i}&\textbf{j}&\textbf{k}\\1&3&-2\\2&-1&4\end{vmatrix}$
$=\textbf{i}(3\times4-(-1\times-2))-\textbf{j}(1\times4-2\times-2)+\textbf{k}(1\times-1-2\times3)$
$=10\textbf{i}-8\textbf{j}-7\textbf{k}$
$\textbf{w}\times\textbf{v}=\begin{vmatrix}\textbf{i}&\textbf{j}&\textbf{k}\\2&-1&4\\1&3&-2\end{vmatrix}$
$=\textbf{i}(-1\times-2-3\times4)-\textbf{j}(2\times-2-4\times1)+\textbf{k}(2\times3-1\times-1)$
$=-10\textbf{i}+8\textbf{j}+7\textbf{k}$
Therefore, the required vectors are $⟨10,-8,-7⟩$ and $⟨-10,8,7⟩$
