Abstract vector algebra with scalars

Question

I am interested in expanding a vector expression in terms of the scalar parameter, but it doesn't give the expected result

$Assumptions = (ϵ) ∈ Reals;
$Assumptions = (r | k) ∈ Vectors[3, Reals];

A[r_] := Module[{M},
$Assumptions = r ∈ Vectors[3, Reals];
M = r + k*ϵ // Simplify;
Return[M]
];

q[r_] := Module[{M},
$Assumptions = r ∈ Vectors[3, Reals];
M = ϵ*(r\[Cross]A[r]) + ϵ*r // Simplify;
Return[M]
];

q[r]

Coefficient[Expand[TensorReduce[q[r]], ϵ], ϵ, 1]

Coefficient[Expand[TensorReduce[q[r]], ϵ], ϵ, 2]

Edit:

Output of the first Coefficient is

    r + r\[Cross](r + k ϵ)

and should be

   r+ r\[Cross]r (* same as just r *)

and output of the second Coefficient is

   0

and should be

  r\[Cross]k

Answer 1

When you set the value of $Assumptions, the old value of $Assumptions is lost. Instead, use Assuming:

WolframLanguageData["Assuming","PlaintextUsage"]

Assuming[assum, expr] evaluates expr with assum appended to $Assumptions, so that assum is included in the default assumptions used by functions such as Refine, Simplify, and Integrate.

In your case you need to use Assuming when you call TensorReduce. So:

$Assumptions = ϵ∈Reals && (r|k)∈Vectors[3, Reals];

A[r_] := r + k ϵ

q[r_] := ϵ (r \[Cross] A[r]) + ϵ r

expanded = Assuming[rp ∈ Vectors[3, Reals], Expand @ TensorReduce[q[rp]]]

Coefficient[expanded, ϵ, 1]

Coefficient[expanded, ϵ, 2]

rp ϵ - ϵ^2 k \[Cross] rp

rp

-k \[Cross] rp