I’m not sure I understand the question enough to answer. Do you mean something like differential geometry? There, the theory is built upon vectors and covectors (i.e., differential forms) that are associated with tangent spaces and cotangent spaces, respectively. But that is modern differential geometry and not classical geometry.
I was asking for a classical geometry book which is using a treatment using vectors. Usually classical geometry is treated without resorting to vectors.
That is still not using vectors as much. My question is if classical geometric problems can be solved using vectors, using vector concepts for example using cross products for area expressions?
