point vs vector

[AntonioLao]

In two dimensions, math definitions give no considerable differences for a point and a vector. Both can be defined as an ordered pair of real numbers mandn. The pointPis given by (m, n) while the vectorVis given by <m, n>. The former implies the existence of a 2D coordinate system as a frame of reference. The latter is independent of such a system althoughmandnrepresent the two components of a unique vector. The pointPalso implies the existence of an origin given by (0, 0).Vis always free of such origin although <0, 0> represents a unique vector of zero length with infinitely possible directions to choose from.

In a Cartesian coordinate system,mrepresents the abscissa whilenrepresents the ordinate. In a polar coordinate system,mrepresentsRcosqandnrepresentsRsinqwhereRis known as the radius vector and qis the angular distance of Rmeasured counterclockwise from the abscissa of zero ordinate usually called the x-axis. Ifq=90° orp/4 thenRcoincides with the vertical y-axis of zero abscissa. By these definitions, Ris a bound vector with one end fixed at the origin (0, 0). On the other hand, <m, n> remains a free vector with its transformed direction given as the ratio ofn/mor tanqwhich is also known as the slope of any line parallel toR.

In one dimension,Pis simply represented by (m) whileVis represented by <m>. Both could imply directional invariance. However, (-m) and <-m> could be used to represent respectively linear reflection or 180° rotation ofPand V. Moreover, <-m> can be used to represent the reverse direction of time. In three dimensions,Pis (m, n, w) andVis <m, n, w>. In four dimensions,Pis (m, n, w, u) andVis <m, n, w, u>. These higher dimensional representations give complicated directional orientations, making the use of a well defined coordinate system absolutely necessary as required by a principle of covariance.

[mkirkpatrick]

Is this leading to a cold fusional conclusion?



regards michael.

[AntonioLao]

The immediate step is a time reversed resolution.

[mkirkpatrick]

The immediate step is a time reversed resolution.

Thats what you are working on here?



regards michael.

[AntonioLao]

Looking back the 3 neutrinos were at the same space-time location. Creating the same space-time location is the same as bringing them back to the same location.

[mkirkpatrick]

Looking back the 3 neutrinos were at the same space-time location. Creating the same space-time location is the same as bringing them back to the same location.

Not easily done!


regards michael.

[AntonioLao]

But the key concept is to accept the topological connectivity of all space-time points.

[mkirkpatrick]

But the key concept is to accept the topological connectivity of all space-time points.

Acceptence is the key then?



regards michael.

[AntonioLao]

That all space-time points are connected although each point is quantized.

[mkirkpatrick]

That all space-time points are connected although each point is quantized.

Yes,that's about the size of it.




regards michael.