Write each vector as x- and y-component vectors and add the component vectors.
Any vector can be broken down into a sum of two other vectors. Breaking the vectors you want to add into its horizontal and vertical components (x and y) is an easy convention to use to make the addition simpler. This guarantees that you will have components that are either in the same direction or perpendicular.
Let's say we want to add
Find the component vectors
Then, to add the vectors, you simply add the magnitudes of the x-components (or subtract if they are in opposite directions to each other) and the magnitudes of the y-components to find the components of the resultant vector.
You can then use the Pythagorean Theorem to find the magnitude of the resultant vector and trigonometry (
If you have them in component form and using the same basis, you just add the components. For example, assume that you are in 2-D Cartesian system, then for:
...we say that:
You "just" add them. Is your thing more complicated?