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MathGroup Archive 1998

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Re: Re: Error in CrossProduct...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14441] Re: Re: Error in CrossProduct...
  • From: Eric Strobel <EStrobel at schafercorp.com>
  • Date: Wed, 21 Oct 1998 03:32:37 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

Hi all,

I thought I'd chime in to clear up something which might have been
confusing

to some students who may 'lurk' on this group...

Dave Withoff wrote:

>
> SNIP...
>
>If you think back to elementary vector analysis, using vectors to
>specify points in space, with nothing said about vector fields, a
>vector from the origin of the space to the point with cartesian
>coordinates x,y,z could be specified by a list of three numbers,
>{x,y,z}.  If spherical coordinates r,theta,phi are used for that point
>in space, then the same vector can be specified by the list
>{r,theta,phi}.
>
>The r in this list can be described as "the r-component of the vector."
>In answer to your first question, this is the "length" (magnitude) of
>the vector.  If this component is zero, the vector is zero.  The other
>two components are just direction angles.
>
>A second meaning of "the r-component of a vector" comes up in describing
>vector fields.  If points in space are specified using spherical
>coordinates, it is customary to introduce a set of three unit vectors
>at each point in space, with one unit vector pointing in the direction
>of increasing r, one pointing in the direction of increasing theta, and
>one pointing in the direction of increasing phi.  The vectors of the
>vector field are then specified by giving their components with respect
>to those three unit vectors.  Within this coordinate system, "the
>r-component of the vector" is not the magnitude of the vector.  It is
>the component of the vector field in the direction of increasing r at
>that particular point.
>
> SNIP...
>

While the entire post was essentially correct and covered what I believe
to  be the core issue, the above paragraphs stood out to me, the second
paragraph 
above, in particular.  The 'r' in the list *may* be described as the "r-
component of the vector", but it isn't really correct to do so.  It is
simply 
the *r-coordinate* !  The only vector this is the magnitude of is the 
coordinate vector of a point in space.  In practice (at least as far as
I've

ever experienced) only the second meaning (see third included paragraph)
is  ever used.

To illustrate: let rr, th, ph be the unit vectors in the r, theta, and
phi  directions.  A vector in spherical coordinates is then...
    V(r, theta, phi) = R(r,theta,phi) rr  +  T(r, theta, phi) th  +
                       P(r,theta,phi) ph The *r-coordinate* is 'r' and
the *r-component* is 'R'.

As was described in Dave's post, Mathematica has no problem computing
cross- products for this, more usual, case.  AND, Mathematica also
correctly handles 
the far more unusual case...

- Eric Strobel
estrobel at schafercorp.com


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