# Solving equations with symbolic variables

I have the equations

eq1=(a - a) \[Psi] . \[Xi] + (a + 2 a) \[Psi] . \[Phi]
eq2=2 (a - a) \[Psi] . \[Xi]
eq3=(a + a + a + a) \[Psi] . \[Phi]


Question: How do I make Mathematica to solve for the coefficients a, a, a and a the system of equations given by eq1=0, eq2=0 and eq3=0?

Elaborating on the question:

When equating any of the above equations to zero, each term should vanish separately, so that eq1=0 gives a - a=0 and a + 2 a=0.

For example, I could solve manually the system eq1=0, eq2=0 and eq3=0 in terms of a, a and a

Solve[{a - a==0,a + 2 a==0,a + a + a + a==0},{a,a,a}]


To obtain

{{a -> -4a, a -> 2a, a -> a}}

But I want Mathematica to do the solving step for me and I don't want to specify for what variables it should solve for, It doesn't matter if It solves for a, a, a as a function of a instead. In my original code I can have more than three independent equations for the coefficients (these are the equations given by imposing eq1=0, eq2=0 and eq3=0) and the number of coefficients is not limited to four. This makes manually solving not feasible.

\$Version

(* "13.0.0 for Mac OS X x86 (64-bit) (December 3, 2021)" *)

Clear["Global*"]

eq1 = (a - a) ψ . ξ + (a + 2 a) ψ . ϕ;
eq2 = 2 (a - a) ψ . ξ;
eq3 = (a + a + a + a) ψ . ϕ;

var = Variables[Level[{eq1, eq2, eq3}, {-1}]]

(* {ξ, ϕ, ψ} *)

sol = Select[Solve[{eq1, eq2, eq3} == 0],
FreeQ[#, Alternatives @@ var] &]

(* {{a -> -4 a, a -> 2 a, a -> a}} *)
`