How do you add vectors in two dimensions

Explanation:

Use the magnitude and direction of each vector to find its components.  Add the components that are along the same dimension.  Then use Pythagorean theorem and trigonometry to find the magnitude and direction of the resultant vector.

For example, if we have a vector of magnitude A and direction α, and another vector of magnitude B and direction β, then the components of the first vector are:

Ax = A cos α

Ay = A sin α

And the components of the second vector are:

Bx = B cos β

By = B sin β

The resultant vector (we'll call it C) has components:

Cx = Ax + Bx

Cy = Ay + By

The magnitude of the resultant vector is:

C = √(Cx² + Cy²)

And the direction of the resultant vector is:

θ = atan(Cy/Cx)