tensors - Problem verifying expression with 3D vectors

I am unable to verify that my vector expressions are equivalent. I want it to say true or false.

 Remove["Global`*"]

$Assumptions = e ∈ Vectors[3, Reals]
$Assumptions = a ∈ Vectors[3, Reals]
$Assumptions = p ∈ Vectors[3, Reals]

p = a - (a.e) e
TensorExpand[p.p] == (a.a - a.((a.e) e) - ((a.e) e).a + ((a.e) e).((a.e) e))
(a.a - a.((a.e) e) - ((a.e) e).a + ((a.e) e).((a.e) e)) == ((a - \
(a.e) e).(a - (a.e) e))
TensorExpand[p.p] == Norm[a]^2 - (a.e)^2 (*e has length 1*)

What I ultimately want is to find (using mathematica) is the angle between vectors a and b. I'm given the lengths of a and b as 156 and Dot[a,e]== Dot[b,e]==90. vectors p and q are the projections of a and b onto a plane. Angle between p and q is 120. Now I have this code that is no good.

   $Assumptions = (a | e) \[Element] Vectors[3, Reals] && e.e == 1 && 
   a.e == b.e == 90  && Sqrt[a.a] == 156 == Sqrt[b.b]; 
   p = a - (a.e) e;

   q = b - Dot[b, e] e;
   Simplify[TensorExpand[a.b], Sqrt[a.a] == 156 == Sqrt[b.b]]