Taylor expansion of function of vectors—simplifying the output

Question

I've got a real-valued function of several vectors $f(u,v,w)$ formed by taking scalar products of linear combinations of the vectors,

I want to Taylor expand around small $v$ by writing

$$f(u,\delta v, w) = A + B\delta + O(\delta^2)$$

for small real $\delta$. When I do this in Mathematica, the output gives me terms like $v(1.u)$ rather than simplifying this to $v.u$.

Note that I have put into my assumptions that $u,v,w$ are all vectors. My Mathematica code is below.

$Assumptions = (p|q|r|P|m|n) ∈ Vectors[4]

(p|q|r|P|m|n) ∈ Vectors[4, Complexes]

FullSimplify[
  TensorExpand[
    Series[((n + q).(n + q)(p + q + δ r).(p + q + δ r))/
     (P.P(p + δ r).(p + δ r) - 
       (p + n + δ r).(p + n + δ r)(p + q + δ r).(p + q + δ r)),
     {δ,0,1}]],
  {p.p == 0, q.q == 0, r.r == 0, n.n == 0, m.m == 0, δ ∈ Reals}]

-(n.q/n.p) + 
  (r n.q (2 (1.n + 1.p + n.1 + p.1) p.q - (1.p + p.1) P.P) δ)/(4 (n.p)^2 p.q) + O[δ]^2

Answer 1

Since it solved the problem, I'll promote my comment to an "official" answer.

To get the behaviour that you require you need to put the TensorExpand inside the Series thus:

$Assumptions = (p | q | r | P | m | n) \[Element] Vectors[4]

(* (p | q | r | P | m | n) \[Element] Vectors[4, Complexes] *)

FullSimplify[
  Series[TensorExpand[((n + q).(n + q) (p + q + \[Delta] r).(p + 
     q + \[Delta] r))/(P.P (p + \[Delta] r).(p + \[Delta] r) - (p \
     + n + \[Delta] r).(p + n + \[Delta] r) (p + q + \[Delta] r).(p + 
     q + \[Delta] r))], {\[Delta], 0, 1}], {p.p == 0, q.q == 0, 
     r.r == 0, n.n == 0, m.m == 0, \[Delta] \[Element] Reals}]

(* SeriesData[\[Delta], 0, {-Dot[n, p]^(-1) Dot[n, q], 
Rational[1, 2] Dot[n, p]^(-2) Dot[n, q] Dot[p, q]^(-1) (
2 Dot[p, q] (Dot[n, r] + Dot[p, r]) - Dot[p, r] Dot[P, P])}, 0, 2, 1] *)