We have already gone over vector decomposition when adding vector together mathmaticaly.  But here is a Video from my idol Dr Walter Lewin from MIT.  

 
Vector Decomposition

The process of finding a vector’s components is known as “resolving,” “decomposing,” or “breaking down” a vector. Let’s take the example, illustrated to the left, of a vector, A, with a magnitude of A and a direction above the x-axis. Because , , and A form a right triangle, we can use trigonometry to solve this problem. Applying the trigonometric definitions of cosine and sine,

Definition provided by Sparknotes 

When dealing with Initial motion is in two directions (at an angle) the first thing you need to do is decompose the velocity in to x direction and y direction.  This needs to done because the pull of gravity is still in the y direction and the initial velocity is effected in the y direction.  The x component of the initial velocity is still not effected by the pull in the y. 

Using the definitions of Cosine and Sine and solving for Ax and Ay we end up with the following Equations