Symbolically associate vectors and their norms

Question

I am wondering how to handle the following situation: I do have vectors of known dimension that I would like to handle symbolically. I suppose I can do something like

avec = Array[a,2];
bvec = Array[b,2];

Now in my equations there are also scalars that represent the norms of these vectors, i.e. A == Norm[avec] So suppose I write an expression like

expr = A avec.(avec + bvec)

What I would like Mathematica to do is:

1. Whenever avec.avec is encountered, replace this by A^2
2. Whenever avec.bvec is encountered, leave this symbolic, don't expand into the elements of the vectors.

How can I achieve this?

Answer 1

One possibility is to replace Plus[] and Dot[] with generic operators like CirclePlus[] (⊕) and CircleTimes[] (⊗), which enables you to do things like

(Distribute[avec⊗(avec⊕bvec), CirclePlus] /. avec⊗avec :> A) /.
 Thread[{avec, bvec} -> {HoldForm[avec], HoldForm[bvec]}]

which yields A⊕avec⊗bvec. The HoldForm[] prevents avec and bvec from being turned into their corresponding values.

Answer 2

Perhaps something like:

ClearAll[symblNm];
SetAttributes[symblNm, HoldFirst];
symblNm[x_] := SymbolName[Unevaluated[x]]

Distribute[symblNm[avec].(symblNm[avec] + symblNm[bvec])]/. {Dot[x_, x_]:> Norm[x]}
(* "avec"."bvec" + Norm["avec"] *)

To evaluate:

Map[Symbol, %, {-1}]
(* Sqrt[Abs[a[1]]^2 + Abs[a[2]]^2] + a[1] b[1] + a[2] b[2] *)

or

Map[ToExpression, %%, {-1}]
(*  Sqrt[Abs[a[1]]^2 + Abs[a[2]]^2] + a[1] b[1] + a[2] b[2] *)