For CG (computer graphics) we use hierarchical coordinate systems. Coordinate systems often called a space (World, Screen/View, Local space) define an origin, scale and rotation in 2D, 3D, ... nD.
Hierarchy connects the coordinate spaces (generally via transformation matrices)
An object in its local space is transformed to world space (places it in the world) then transformed to screen space (puts it on the screen).
You must select which coordinate space you want a vector to work in. If you have a vector in local space and want to add a vector from view space you must transform one of the vectors to the others space.
Most game frameworks will come with all the utilities needed to manipulate vectors, matrices, and coordinate spaces. However I would strongly suggest that you write your own so you can get a thorough feel of what is going on. If you are comfortable using vectors then it will not be that difficult with a bit of research.
"If I scale one object to be large enough to overlap/collide with another object does that mean those two objects are still in their own subspaces?"
Yes until you transform them both to world space. They will then be in the same coordinate system. You can then test if they overlap.