Re: Simple books on 4-vectors

From: Danny Ross Lunsford (antimatter33_at_yahoo.com)
Date: 10/13/04 

Date: Wed, 13 Oct 2004 08:45:20 +0000 (UTC)

Oz <oz@farmeroz.port995.com> wrote in message news:<xb2zOWA22oZBFww8@farmeroz.port995.com>...

> >I assume you are familiar with the notion of a vector. Let's defer the
> >precise definition until later. It's a "directed length" - which
> >means, it has a direction and a length. Let's call it an "oriented
> >length" - oriented because you pick the direction from one endpoint to
> >the other. Let's call it
> >
> >X
>
> OK.
> As a detail, can we not say that in principle we can express this object
> as being in any number of dimensions by re-orienting any arbitrary set
> of axes so one 'points' in the direction of X. In this case X can be
> expressed as two numbers, 'start' and 'end' on some number line. We note
> that
>
> end - start = - (start - end)
>
> Direction in this case is either + or - so is a 0D dimension.

Don't worry about dimensions yet, only that eventually we can't make
anything beyond an N-form.

> Oooh. Trickier.
>
> >The magnitude of X ^ Y is just the area of the parallelogram.
>
> Hang on. You haven't defined 'area'.
> It certainly isn't typically a linear combination of X & Y.

Whoa - who said anything about linear combinations? Stick to the plan.
I haven't said a word about how these things transform yet. So far all
we've done is take your intuitive idea of a vector as an "oriented
length" and shown a way to "oriented N-lengths".

> >Now suppose X and Y are the same oriented length X = Y. The
> >parallelogram collapses to a line segment and the area is 0. That is,
> >
> >X ^ Y = 0 if X = Y
>
> Yes, but hang on. What you are really saying ix X ^ X = 0
> Ahh, hang on, you are saying nX ^ X = 0, n any real number.

No one said a word about components or real numbers yet, other than
the magnitude.

> Ahh, but we should be able to define an orthogonal axis this way.
> Hmm, needs some thought though.

No one said a word about orthogonal transformations yet.

So let's back up - so far we have

X - oriented length - vector

X^Y - oriented surface element - bivector

X^Y^Z - oriented volume element - trivector

X^Y^Z^W - in 3D, 0.

Clear? Next we'll talk about linearity and I'll be more precise about
magnitude, after introducing a few more rules.

-drl

Relevant Pages

  • Re: Is a point a vector?
    ... a magnitude and a direction but the point does not. ... but there are points not only in affine spaces ... vectors and linear forms to be the *same* object) ... At least that's my opinion. ...
    (sci.math)
  • Re: Is a point a vector?
    ... a magnitude and a direction but the point does not. ... but there are points not only in affine spaces ... vectors and linear forms to be the *same* object) ... At least that's my opinion. ...
    (sci.math)
  • Re: Is a point a vector?
    ... then any other point on that line will have a magnitude and direction. ... but there are points not only in affine spaces ... one can establish a correspondence between vectors in a vector ... vectors and linear forms to be the *same* object) ...
    (sci.math)
  • Re: Logarithmic Interpolation
    ... ....and since the shifted logfreq sequence will be complex, ... take the magnitude again IF you intend to move it all back to a linear ...
    (comp.dsp)