Re: components

["John S. Denker" <jsd@MONMOUTH.COM>]

Joe Heafner wrote:
> I hope this is a sensible question.
>
> Does it make sense to speak of the components of a
> vector before defining a coordinate system?

Well, the question is a bit tricky.

1) To answer a slightly different question:  No, it does
not make sense to speak of the components of a vector
before defining a _basis_.  Proof: The choice of basis
is arbitrary.  The value of the components depends on
the choice of basis.  Therefore if you don't know the
basis, anything you say about an individual component
is nonsense.

2) You can say sensible things about certain special
_combinations_ of components, such as the sum of the
squares of the components, if you can prove the combination
is independent of the choice of basis.

3) Technically, defining a basis is not quite the same
as defining a coordinate system.  The distinction becomes
significant when dealing with a curved space.  Imagine
a point somewhere in Greenland.  You can form a basis
consisting of a Northward unit vector, and Eastward
unit vector, and an Upward unit vector.  These will
_not_ coincide with unit displacements along the
principal directions of the standard spherical
coordinate system.  The basis vectors are straight
but the coordinate system is curved.