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