Remove Abs from Norms of Vectors

expr = Norm[{a, b*c}]

Sqrt[Abs[a]^2 + Abs[b c]^2]

Since ComplexExpand assumes all its variables to be real, we automatically get what we want.

ComplexExpand@expr

Sqrt[a^2 + b^2 c^2]

Refine[expr, {a > 0, b c > 0}]

Sqrt[a^2 + b^2 c^2]

and

FunctionExpand[expr, {a > 0, b c > 0}]

Sqrt[a^2 + b^2 c^2]

If you have to use FullSimplify or Simplify, you can use the option ComplexityFunction to make expressions with Abs more costly:

FullSimplify[Norm[{a, b*c}], Assumptions -> {a > 0, b > 0, c > 0}, 
  ComplexityFunction -> (100 Count[#, _Abs, {0, Infinity}] +  LeafCount[#] &)]

Sqrt[a^2 + b^2 c^2]

Also, for a real number x, Abs[x] = Sqrt[x^2]

Norm[{a, b*c}] /. Abs[x_] :> Sqrt[x^2]

(* Sqrt[a^2 + b^2 c^2] *)

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