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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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