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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


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