I have two points: the origin $O(0, 0)$ and a point $A(11, -12)$.
There is an unknown line segment also connected to $A$ (drawn in red above) and I know the angle between the red and yellow lines which is $\theta=140^\circ$.
I would like to draw the red vector starting from $A$. How do I find the equation of that vector?
Rotation about the origin by angle $\theta$ counterclockwise in the plane is given by the linear transformation $$\begin{bmatrix}\cos \theta & -\sin \theta \\ \sin\theta & \cos \theta\end{bmatrix}.$$ (If like me you get confused about which off-diagonal term has the minus sign, try $\theta=90^{\circ}$ and test it on a vector, like $(1,0)$).
So now take the vector from $A$ to $O$, rotate it, and that gives you the direction of the red line.
If you need more help, post what you get so far and where you get stuck.