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Re: Dot product confusion

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109956] Re: Dot product confusion
  • From: Barrie Stokes <Barrie.Stokes at newcastle.edu.au>
  • Date: Wed, 26 May 2010 07:09:39 -0400 (EDT)

Hi Steve

>From the Mathematica documentation for Dot:
"The dimensions of the result are those of the input with the common dimension collapsed:"
There is then an example where an object with Dimensions "{2, 3, 4}" Dotted with an object with Dimensions "{4,5,2}" gives an object with dimensions "{2,3,5,2}".

ptsa = {{x1, y1, z1}, {x2, y2, z2}, {x3, y3, z3}}
Dimensions @ ptsa

The matrix ptsa has dimensions 3*3.

Dimensions @ {aa, bb, cc}

{aa, bb, cc} is a list with dimension 3:

The product {aa, bb, cc}.ptsa is thus the dot product  of a "3" and a "3*3" giving a result with dimension 3:

{aa, bb, cc}.ptsa  (* expression 1 *)
Dimensions @ %

On the other hand, ptsa.{aa, bb, cc}  is the dot product of  a " 3*3" and a "3"  giving (another) result with dimension 3:

ptsa.{aa, bb, cc}  (* expression 2 *)
Dimensions @ %

Note that:
Dimensions @ Transpose[{{aa, bb, cc}} ]

Transpose[{{aa, bb, cc}} ] is a column vector with dimensions 3*1.

So, ptsa.Transpose[{{aa, bb, cc}} ] is the dot product of  a " 3*3" and a "3*1"  giving a result with dimension 3*1:

ptsa.Transpose[{{aa, bb, cc}} ]  (* expression 3 *)
Dimensions @ %

Barrie

>>> On 25/05/2010 at 8:32 pm, in message <201005251032.GAA20697 at smc.vnet.net>, "S.
B. Gray" <stevebg at ROADRUNNER.COM> wrote:
> 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.
> 
> Steve Gray



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