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Symbolic vector handling

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83353] Symbolic vector handling
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Sun, 18 Nov 2007 04:46:03 -0500 (EST)

Hello group,

this is most probably a very simple question but still ...

Consider a set of vectors vA, vB, vC, ... and scalars a, b, c, ...

What I want to do is a symbolic handling of expressions like vA.(b vB + 
vC)
which should be expanded to

(1)    expr = vA.(b vB + vC) -> b vA.vB + vA.vC

i.e. I want Mathematica to apply the distributive law and the 
extraction of scalars.
I don't want to use any coordinate representation.

The first thing that comes into mind would be to define a type "vector"
and another type "scalar" and then define the usual rules.

I'd like to execute a comand like

    FunctionExpand[ expr , Assumptions->{{vA, vB, vC} "elem" vectors, 
{a,b,c} "elem" scalars}]

Here's an example of a simple problem in which my question arises:

Let the vectors vR1 and vR2 be defined in terms of four other vectors 
vA1, vA2, vN1, vN2 and two scalars a and b as follows:

vR1 = vA1 + a vN1
vR2 = vA2 + b vN2

and consider the function

U = (vR1 - vR2).(vR1 - vR2)

The task is to find the minimum of U in terms of a combination of 
scalar product of the vectors vA1, vA2, vN1, vN2 and of the optimizing 
parameters a and b.

Any help is greatly appreciated.

Regards,
Wolfgang 



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