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From: Vince Virgilio on 26 May 2010 07:06
On May 25, 6:32 am, "S. B. Gray" <stevebg(a)ROADRUNNER.COM> wrote:

SNIP

> ambiguous, but I assumed the dot product would always commute. Should
> there be a warning?

SNIP

Nope, this is just the usual behavior of an inner product of a matrix
and vector.

The doc. says that Dot[] contracts the last dimension of the first
tensor, with the first dimension of the second tensor. So if either
operand has dims > 1, Dot[] doesn't commute.

Vince Virgilio

From: Simon on 26 May 2010 07:08
Hi Steve,

The result is correct, since {aa,bb,cc} is treated as a vector that
can be either row or column.
Then, thinking of matrix multiplication - with indices and summation
convention
(A.v)_i = A_{i j} v_j => A.v = ( A_{1 j}v_j , A_{2 j}v_j , A_{3 j}
v_j )
(v.A)_i = v_j A_{j i} => v.A = ( A_{j 1}v_j , A_{j 2}v_j , A_{j 3 }
v_j )

If you want your dot products to be unambiguous, then write your
vector as a 3x1 or a 1x3 matrix - equivalent to a column and row
vector resp.
So using your objects:

In[1]:= ptsa={{x1,y1,z1},{x2,y2,z2},{x3,y3,z3}};

Column vector:
In[2]:= ptsa.{{aa},{bb},{cc}}
Out[2]= {{aa x1+bb y1+cc z1},{aa x2+bb y2+cc z2},{aa x3+bb y3+cc z3}}

Fails if used the wrong way round:
In[3]:= {{aa},{bb},{cc}}.ptsa
During evaluation of In[3]:= Dot::dotsh: Tensors {{aa},{bb},{cc}} and
{{x1,y1,z1},{x2,y2,z2},{x3,y3,z3}} have incompatible shapes. >>
Out[3]= {{aa},{bb},{cc}}.{{x1,y1,z1},{x2,y2,z2},{x3,y3,z3}}

Row vector:
In[4]:= {{aa,bb,cc}}.ptsa
Out[4]= {{aa x1+bb x2+cc x3,aa y1+bb y2+cc y3,aa z1+bb z2+cc z3}}

Fails if used the wrong way around:
In[5]:= ptsa.{{aa,bb,cc}}
During evaluation of In[5]:= Dot::dotsh: Tensors {{x1,y1,z1},
{x2,y2,z2},{x3,y3,z3}} and {{aa,bb,cc}} have incompatible shapes. >>
Out[5]= {{x1,y1,z1},{x2,y2,z2},{x3,y3,z3}}.{{aa,bb,cc}}

~~~
Simon

On May 25, 6:32 pm, "S. B. Gray" <stev...(a)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


From: inOr on 26 May 2010 07:08
On May 25, 3:32 am, "S. B. Gray" <stev...(a)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

Steve,
The dot product is most definitely NON-commutative. To see this for a
case like yours, think of vector-matrix multiplication as the
transformation of a vector into a new vector. In real number
representation, pre-multiplication of a vector by a matrix creates a
new vector that is the sum of the column vectors of the matrix
weighted by the elements of the vector. In this case, the matrix
multiplication creates a column vector out of another column vector.
When the vector is post-multiplied by the matrix, the result is a
vector that is the sum of the ROW vectors of the matrix, still
weighted by the components of the vector. In this case, a row vector
is transformed into another row vector. Disregarding the row- /
column- difference, the two results are equal only if the rows and
columns of the matrix are identical (symmetric matrix).
Mark Harder

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