we think of a point as a spot in an Euclidean plane
indicating a position.Intuitively we assign arbitrary
framework.The confusion (for me) is when
we consider a line segment.It is made of
infinite points aligned in a certain direction.
So a segment results when all these “ directions”
are added vectorial, but a zero vector is
conceptually an infinite direction entity
so how is this vector direction specified for
a point nathematically. physically of course
we have a visible dot with visible dimension,
Any insight or explanation from all sources
is welcome.
I am not sure if I really understand your question but:
A point is not a direction or a vector.
A Point is an element of the euclidean space which can be represented by its position vector.
A line segment is a subset of all points which match a specific rule concerning the position vectors.
I hope that helps.
If not, please give some more details of your question.
Volker
Dear Volker Böß Thank you for your response.Yes, that is my view
the abstraction of a point would be a position vector, and I agree
that a segment or any curve could be represented by a rule
on the position vector for example say y= mx+ c but what to my
mind is puzzling is this implies a pre-defined co- ordinate system
while in Euclidean geometry we do not specify any frame.
Actually the point arose in my mind as I toyed with the idea
of representing Euclid geometry completely in matrix( linear algebra)
form. I am not a mathematician, if the question is better explained
now, would be delighted to know if my thinking makes sense
at all.
Cheers
Dear Volker Böß Thank you for your response.Yes, that is my view
the abstraction of a point would be a position vector, and I agree
that a segment or any curve could be represented by a rule
on the position vector for example say y= mx+ c but what to my
mind is puzzling is this implies a pre-defined co- ordinate system
while in Euclidean geometry we do not specify any frame.
Actually the point arose in my mind as I toyed with the idea
of representing Euclid geometry completely in matrix( linear algebra)
form. I am not a mathematician, if the question is better explained
now, would be delighted to know if my thinking makes sense
at all.
Cheers
So you mean that you want to define a coordinate system in a given euclidean plane or space (which fullfills the axioms of an affine plane)?
To do this you need to select a point and two non collinear vectors which define the origin O and the vectors e1 (1,0) and e2 (0,1). The position of a point can then be defined by a linear combination of 0+a*e1+b*e2, which is (a,b).
If you choose another coordinatesystem for the same plane the coordinates will be different.
For a general descriptions of curves you should use the form P(t)=( x(t), y(t) ) with t as an interval and continuous functions x and y.
Dear Volker Böß thank you for your kind informative response.
Yes, vector space in liner algebra is well defined and always makes displacement as a means for adding structure and matrix
theory uses homogeneous coordinates for affine transformation. Now if we view a vector( with parameter t to represent a line) then
in this picture , it could be taken as a transformation
of the real line.Real lines have irrational “ points”( open
to correction) so a line would have I think
deep infinite holes in this picture. Somewhere
it appears that we have to abandon the position
vector also and stick to modern abstract vector space,
abandoning Euclid. Just as an example of trying
to use linear algebra , e.g. we can view a 3x3 matrix
as a triangle determined by each column vector.Then
we could define the median points by applying a transformation
matrix with( .5 -.5 0) (0 .5 -.5 ) ( -.5 0 .5) vectors.
so can we now derive properties of this matrix
to show concurrency?
Cheers
Continuing the concept of zero vector, inputs on some doubts
below would be very welcome.
1. is a zero vector in a Descartes plane an infinite dimensional
vector. The intuition being , a vector basically indicates direction
even in linear algebra.Every vector in L.A. is always interpreted
as a position vector( displacement? ) from the origin thereby
indicating direction from origin.
2.So if we assume that the origin zero has all 360 degrees of freedom ( as a continuum ) then we have perfect symmetry of infinite dimension wrt
to direction( would this be correct?). Thus any non zero
vector would indicate that a particular choice has been made.
This contrast with abstraction of vector space where zero
is just an identity related to a composition.
Would welcome criticism of these thoughts.
Cheers
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