Re: Symbolic vector handling

*To*: mathgroup at smc.vnet.net*Subject*: [mg83408] Re: [mg83353] Symbolic vector handling*From*: danl at wolfram.com*Date*: Mon, 19 Nov 2007 06:23:51 -0500 (EST)*References*: <200711180946.EAA01450@smc.vnet.net>

> 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 There are probably a few reasonable ways to go about this. One is to use a special vector "head" and define rules for handling distribution involving Plus, Times, and whatever else might be of relevance. I encapsulate your vectors with vec[]. I find it convenient to move to a new function, myDot, for purposes of intermediate computation. This avoids placing new behavior on Dot itself. myDot[a_] /; Head[a] =!= vec := a myDot[x___, y_Plus, z___] := Map[myDot[x, #, z] &, y] myDot[x___, y_Dot, z___] := myDot[x, Apply[Sequence, y], z] myDot[x___, y_Times, z___] := myDot[x, Apply[Sequence, y], z] myDot[x___, y_myDot, z___] := myDot[x, Apply[Sequence, y], z] myDot[] = 1; myDot[x___, y_, z___] /; Head[y] =!= vec := y*myDot[x, z] DotExpand[expr_] := (expr /. Dot -> myDot) /. myDot -> Dot In[54]:= DotExpand[vec[vA].(b*vec[vB] + vec[vC])] Out[54]= b vec[vA].vec[vB] + vec[vA].vec[vC] The above myDot is an alteration of some code I cribbed from the section "Some noncommutative algebraic manipulation" in the 1998 conference notebook at: http://library.wolfram.com/infocenter/Conferences/325/ Daniel Lichtblau Wolfram Research

**References**:**Symbolic vector handling***From:*"Dr. Wolfgang Hintze" <weh@snafu.de>