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Topic: What does one call vector geometry without a coordinate system?
Replies: 32   Last Post: Sep 9, 2013 4:45 PM

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 Ken.Pledger@vuw.ac.nz Posts: 1,332 Registered: 12/3/04
Re: What does one call vector geometry without a coordinate system?
Posted: Sep 1, 2013 5:05 PM

lite.on.beta@gmail.com 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.

Ken Pledger.