Lagrangian mechanics - constraint forces & virtual work

The constraint forces have the dot product 0 ($\textbf C \cdot \textbf x=0$, where $\textbf C$ and $\textbf x$ are vectors, x being the virtual displacement. But the dot product is 0 if $\textbf C$ and $\textbf x$ are perpendicular. So, are the constraint forces always perpendicular to the virtual displacement? Forces such as tension are not always perpendicular to the virtual displacement. Thus, I am asking why forces like tension are not written in the lagrangian equation of lagrangian mechanics? To be more specific, see the example of the Atwood machine.