Q:

Why is it okay to decompose forces?

I guess the title says it all, it doesn't seem intuitive that we can consider something as abstract as forces to be decomposable vectors. An example would be a mass sliding off an inclined plane, we consider the tangential component of the weight to the plane. What guarantees us in this case that our results will be correct?

A:

It is clear that displacement is a vector. Since displacement is a vector, so is its derivative with respect to time, which we call velocity. Momentum is velocity times mass, which is a scalar - so momentum is also a vector.
Newton's second law (justified by experimental evidence) says that force is equal to the rate of change of momentum with respect to time. Since momentum is a vector, then force is also a vector. And this is what guarantees that we get the same answer whether we regard a net force as a single item or as the superposition of several components.

