What are the formulas to sum two 2D vectors?

Question

I'm writing a PHP algorithm to calculate the sum of two 2D vectors with the same origin, given their intensity and their angle with a given axis. How to calculate the properties of the sum vector? I have some ideas about the intensity (Pythagorean theorem and Carthesian components), but I don't have a clue of how to calculate the angle.

Answer 1

If your resultant vector is $c$ and $\hat c = c/|c|$, then $\hat c \cdot \hat x = \cos \theta$ and $\hat c \cdot \hat y = \sin \theta$, where $\hat x, \hat y$ are the unit vectors in the x- and y-directions and $\theta$ is the angle $c$ makes with respect to the x-axis.