|MadSci Network: Astronomy|
A big question, which is difficult to answer shortly. Like all of modern physics, vector algebra underlies most of astrophysics. Examples are numerous; for example, to model electromagnetic radiation - light - we need a vector field and the ability to do vector calculus. General relativity uses tensors, which are an extrapolation of vectors and use much of the same algebra. In short, you can't get away from vectors, although I wasn't able to come up with any use for them in planetary geology (although I'm sure I'm just missing something)
Try the links in the MadSci Library for more information on Astronomy.