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# What a vector is.

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Greetings. Please correct me if I am wrong but: A vector is a member of a vector space, with all the attendant axioms and theorems. Vectors are represented as (x, y, z) or(bad ascii art for nx1 matrix) [x] [y] [z] Now, vectors can be converted to a angle/magnitude system. In my learning about graphics and math, both in classes and out of, vectors are usually represented as a nx1 matrix. The only time when I''ve used angle/mag representations is in trig. Why are angle/mag representations commonly defined to be a vector(other than they can be converted to be nx1 matrices) ? ~V''lion Bugle4d

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Hi,
it all depends on the application.

If you are working in a polar coordinate system, you must use the angle/mag notation.
Another example is the math of complex numbers. These numbers can also be described using magnitude and angle.

Both forms are equivalent, they just exist to fit specific needs, depending on the job you are doing.

Hope this helps,
Gerd

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The specific representation of a vector is not relevant. The geometrical vectors are all part of the same vector space regardless of if they are written in cartesian, spherical or cylindrical coordinates.

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quote:
Original post by birger
The specific representation of a vector is not relevant. The geometrical vectors are all part of the same vector space regardless of if they are written in cartesian, spherical or cylindrical coordinates.

That''s not quite true. Cartesian, spherical, and cylindrical coordinates are in fact different spaces. They may represent the same physical volume but the measurement system itself defines the space in which vectors are represented.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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> That's not quite true. Cartesian, spherical, and cylindrical
> coordinates are in fact different spaces.

I disagree. A vector space is a space of vectors which satisfy certain conditions, but the coordinate system is not part of the conditions. E.g. Euclidian 3 dimensional space is a real vector space. Cartesian, cylindrical and spherical are three common coordinate systems used with it, but they do not define seperate spaces.

[edited by - johnb on December 10, 2002 12:02:20 PM]

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Actually, cartesian, spherical and cylindrical co-ordinates can all be used to describe the same vector space. The Vector Space axioms as applied to vector space V,+,*,(F,+,*) (where F,+,* is a field and + and * are overloaded) are:

V is non-empty
V is an abelian group under +
For any a,b in F and u,v in V:
(a*v) is in V
a*(u+v)=a*u+a*v
(a+b)*v=a*v+b*v
(a*b)*v=a*(b*v)
1*v=v where 1 is the multiplicative identity in F.

Since the form in which you represent three dimensional vectors doesn''t affect the behaviour of the vectors under the operations used in the axioms (though it does affect the behaviour of the ordered triples used to represent the vectors), changing the representation doesn''t change the vector space.

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Of course, if you define 3D coordinate systems as the spaces: R*R*R, ((0,infinity)*(R/(-pi,pi])*R)union(0,0,R) and ((0,infinity)*(R/(-pi,pi])*(R/(0,pi)))union(R,0,0) (cartesian, cylindrical and spherical repectively) providing the appropriate operations for each (only R*R*R has a naturally derived addition and scalar multiplication) then you get three isomorphic vector spaces - which is effectively the same thing as the three being the same vector space.

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that''s not a vector, that''s a point

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quote:
Original post by johnb
> That''s not quite true. Cartesian, spherical, and cylindrical
> coordinates are in fact different spaces.

I disagree. A vector space is a space of vectors which satisfy certain conditions, but the coordinate system is not part of the conditions. E.g. Euclidian 3 dimensional space is a real vector space. Cartesian, cylindrical and spherical are three common coordinate systems used with it, but they do not define seperate spaces.

[edited by - johnb on December 10, 2002 12:02:20 PM]

DAMMIT! Now why''d y''all have to go and prove me wrong, .

Based on this info provided by johnb and comments by others, I''m in full agreement that different coordinate systems can represent the same space.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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