fom
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Re: What does one call vector geometry without a coordinate system?
Posted:
Aug 31, 2013 10:46 AM


On 8/30/2013 9:25 PM, William Elliot wrote: > On Sat, 31 Aug 2013, Sam Sung wrote: >> On Fri, 30 Aug 2013 16:30:23 0700 (PDT), 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. >> >> They called it be topology... >> > That's wrong. It's a vector space. >
In the kind of geometric situation described here it is more than a vector space.
Vectors in this situation are equivalence classes of directed line segments (magnitude and direction). The original poster is discussing specific, labeled directed line segments.
It would be called a Euclidean point space. The point difference (the additional algebraic structure) then becomes a ground for a distance function.
I posted the definition on Aug 21 2013 in the thread "What is the modern setting for Euclidean Geometry"
Fred Jeffries found an online version of the text from which I obtained the definition here:
<quote>
Professor Bowen has made this book available for download from his web page: http://www1.mengr.tamu.edu/rbowen/
Unfortunately, the download links do not appear to be working (for me) but these do work:
http://ohkawa.cc.ithiroshima.ac.jp/AoPS.pdf/MathTextBook/Introduction%20to%20Vectors%20and%20Tensors%20Vol%201%20%28Bowen%20314%29.pdf
http://ohkawa.cc.ithiroshima.ac.jp/AoPS.pdf/MathTextBook/Introduction%20to%20Vectors%20and%20Tensors%20Vol%202%20%28Bowen%20246%29.pdf
The definition of Euclidean point space is from the first chapter of volume 2.
<endquote>

