In article <email@example.com>, firstname.lastname@example.org wrote:
> I remember in grade school (grade 8 or 9ish) we did geometry with vectors as > a tool but without any coordinate system. > > We often picked a random point and called it O, then proved things like if M > was midpoint between two points A, B, then: > > OM = 1/2 * (OA + OB) > > A lot of stuff was proven with just points and vectors from points to other > points (with the fact OX = -XO used heavily). > > What's the name of this geometry? I'm thinking of looking this stuff up > again.
Other people have mentioned the theoretical _content_, but I think you're really interested in the _method_. Although some mathematicians don't have much experience of it, it's an efficient way of handling mechanics as well as some areas of geometry. I just call it "vector geometry". Some linear algebra textbooks have a chapter on it.
If you stick to linear combinations of vectors (as in your example), then you are doing affine geometry. Euclidean geometry is the special case which also uses dot products of vectors, as these enable you to calculate lengths and angles.