proposal about vectors

According to the recent discussion I propose the following 
basic conception for vectors:

(%i1) load("vectors.lisp")$
(%i2) v: vector(a,b)$
(%i3) w: vector(c,d)$
(%i4) m: matrix([i,j],[p,q])$
(%i5) declare(s, scalar)$
(%i6) display2d: false$
(%i7) stardisp: true$

Implicit simplification:

(%i8) v+v;
(%o8) 2*vector(a,b)
(%i9) v+w;
(%o9) vector(c,d)+vector(a,b)
(%i10) v-v;
(%o10) vector(0,0)
(%i11) 0*v;
(%o11) vector(0,0)
(%i12) v*x;
(%o12) x*vector(a,b)
commutative, non-vector x is displayed left
(%i13) s*v;
(%o13) s*vector(a,b)
(%i14) v.w;
(%o14) vector(a,b) . vector(c,d)
(%i15) m.v;
(%o15) matrix([i,j],[p,q]) . vector(a,b)

Explicit simplification:

(%i16) v+w, vector_simp;
(%o16) vector(c+a,d+b)
(%i17) x*v, vector_simp;
(%o17) x*vector(a,b)
(%i18) s*v, vector_simp;
(%o18) vector(s*a,s*b)
(%i19) v.w, vector_simp;
(%o19) b . d+a . c
(%i20) w.v, vector_simp;
(%o20) d . b+c . a
(%i21) m.v, vector_simp;
(%o21) vector(j . b+i . a,q . b+p . a)
(%i22) v.m, vector_simp;
(%o22) vector(a,b) . matrix([i,j],[p,q])
(%i23) x.v, vector_simp;
(%o23) x . vector(a,b)
(%i24) v.x, vector_simp;
(%o24) vector(a,b) . x
(%i25) declare(a,scalar)$
(%i26) v.w, vector_simp;
(%o26) b . d+a . c
(%i27) dotscrules:true$
(%i28) v.w, vector_simp;
(%o28) a*c+b . d

Reconstruction of multiplicative and additive expressions:

(%i29) vector_rebuild( vector(s*a+c,s*b+d), [s] );
(%o29) vector(c,d)+s*vector(a,b)

This conception uses as few as possible implicit simplification  
and allows to reconstruct results of multiplicative-additive 
computations in a readable form. E.g. a working example:

(%i1) load("vectors.lisp")$
(%i2) alias(vec, vector)$
(%i3) display2d:false$
(%i4) declare([p,q,r,s,u,v],scalar)$
(%i5) plane1: vec(1,2,3)+p*vec(4,5,6)+q*vec(7,8,9);
(%o5) q*vec(7,8,9)+p*vec(4,5,6)+vec(1,2,3)
(%i6) plane2: vec(3,2,1)+r*vec(6,5,4)+s*vec(9,8,7)$
(%i7) veq: rhs(plane1) = rhs(plane2);
(%o7) q*vec(7,8,9)+p*vec(4,5,6)+vec(1,2,3) = s*vec(9,8,7)+r*vec(6,5,4)+vec(3,2,1)
(%i8) map(disp, sys: extract_equations(veq))$
7*q+4*p+1 = 9*s+6*r+3

8*q+5*p+2 = 8*s+5*r+2

9*q+6*p+3 = 7*s+4*r+1

(%i9) params: algsys(sys,[p,q,r,s]);
(%o9) [[p = %r1,q = %r2,r = -1/3*(16*%r2+13*%r1+16),s = 1/3*(13*%r2+10*%r1+10)]]
(%i10) params: params, %r1=u, %r2=v$
(%i11) plane1, params;
(%o11) v*vec(7,8,9)+u*vec(4,5,6)+vec(1,2,3)
(%i12) plane2, params;
(%o12) 1/3*(13*v+10*u+10)*vec(9,8,7)-1/3*(16*v+13*u+16)*vec(6,5,4)+vec(3,2,1)
(%i13) vector_rebuild(plane2,[v,u]), params;
(%o13) v*vec(7,8,9)+u*vec(4,5,6)+vec(1,2,3)


Display properties:

(%i1) load("vectors.lisp")$
(%i2) alias(vec,vector)$
(%i3) v: vec(1,2);
                                     [ 1 ]
(%o3)                                [   ]
                                     [ 2 ]
(%i4) [vectorp(v), matrixp(v), listp(v)];
(%o4)                        [true, false, false]
(%i5) vector_display2d:false$
(%i6) w: apply(vec,[3,4]);
(%o6)                               [3, 4]
(%i7) [vectorp(w), matrixp(w), listp(w)];
(%o7)                        [true, false, false]

alternatively
vector_display2d: col, row or text
with mode col and row like above and text like
(%i3) v: vec(1,2);
(%o3)                             vector(1,2)

Volker van Nek