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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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 fom Posts: 1,968 Registered: 12/4/12
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>

his web page:
http://www1.mengr.tamu.edu/rbowen/

(for me) but these do work:

http://ohkawa.cc.it-hiroshima.ac.jp/AoPS.pdf/MathTextBook/Introduction%20to%20Vectors%20and%20Tensors%20Vol%201%20%28Bowen%20314%29.pdf

http://ohkawa.cc.it-hiroshima.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>