Re: Dot product confusion
- To: mathgroup at smc.vnet.net
- Subject: [mg109963] Re: Dot product confusion
- From: Roland Franzius <roland.franzius at uos.de>
- Date: Wed, 26 May 2010 07:10:56 -0400 (EDT)
- References: <htg90l$k7h$1@smc.vnet.net>
S. B. Gray schrieb:
> Given
>
> ptsa = {{x1, y1, z1}, {x2, y2, z2}, {x3, y3, z3}};
>
> I thought the following expressions would be identical:
>
> {aa, bb, cc}.ptsa (* expression 1 *)
> ptsa.{aa, bb, cc} (* expression 2 *)
>
> but they are not. They evaluate respectively as:
>
> {aa x1 + bb x2 + cc x3, aa y1 + bb y2 + cc y3,
> aa z1 + bb z2 + cc z3}
>
> {aa x1 + bb y1 + cc z1, aa x2 + bb y2 + cc z2,
> aa x3 + bb y3 + cc z3}
>
> Since ptsa is itself three xyz coordinates, the expressions might be
> ambiguous, but I assumed the dot product would always commute. Should
> there be a warning?
>
> The first result is the one I want.
The first result is mathematically correct as a matrix product with a
left factor a (1x3) matrix and the right factor 3x3 matrix. Nevertheless
for working in the index spaces it is better to use {{aa,bb,cc}} for a
row vector
The second product is mathematically incorrect in the context of general
matrix multiplication because a matrix product of 3x3 . 1x3 does not
exist. but it is conveniently introduced for abuse of notation by lazy
physicists.
In the second product the right factor has to be a 3x1 matrix - or a
column vector - {{aa},{bb},{cc}} and the result has to be of the same type.
Try to
Transpose[{aa,bb,bb}]
No such problems with Transpose[{{aa,bb,bb}}]
--
Roland Franzius