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Re: components

At 21:59 -0500 12/13/01, Joe Heafner wrote:

> From: Hugh Haskell <hhaskell@MINDSPRING.COM>

> Is there an ulterior motive for your question?

Nope. But I *am* trying to see if I can summarize an approach to
dealing with vectors that emphasizes their geometric properties,
independent of a coordinate system.

That should be doable. different vectors don't have to be components
of a third one. They don't even have to be related. Two velocities of
different objects can be combined to yield various relative
velocities. Force and displacement can be combined to yield work.
Etc. None of this has to be done in the context of a particular
coordinate system, although it is often more convenient to do so. You
can think about the dot product as the magnitude of the projection of
one vector on the other, divided by the magnitude of the vector
projected upon. The magnitude of a cross product is the area of the
parallelogram formed by the two vectors, and the triple product is
the area of the parallelopiped formed by the three vectors. None of
these things requires that a coordinate system be defined, although
lots of geometric things that are well handled by coordinate systems
are formed by combinations of vectors--two vectors define a plane
(unless they are colinear). If a third no-coplanar vector is added
that defines a three-dimensional space.

I would think what you want to do can be done, but what is it going
to get you? Why not embrace the coordinate systems and show that any
individual vector is independent of coordinate system, although its
components clearly are not, and from this we find out how to
represent vectors in rotating coordinate systems, or vice versa.

Or better yet, show how the geometric properties of vectors are
easily handled by a coordinate system, and because of the invariance
of a vector under changes of coordinate system, showing a geometric
property in one coordinate system shows it in all. That's a result
that I find really satisfying.

Hugh
--

Hugh Haskell
<mailto://haskell@ncssm.edu>
<mailto://hhaskell@mindspring.com>

(919) 467-7610

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