Why do we choose a coordinate system when adding or subtracting vectors?

Vector quantities are defined by their magnitude and directions

Explanation:

When resolve two vectors, say, A and B moving at different speed and collides after some time. We notice the other quantity B will take the direction of the second body A because it moves in a greater speed.

Mathematically,

- A + B

Which clearly shows, the consideration of coordinate when resolving through vector quantities