Using the Norm Function

Question

I am trying to take the norm of a general vector and show that for vectors, v,w, that

Norm[v cross w]^2==Norm[v]^2*Norm[w]^2-(v dot w)^2

In Mathematica, here are my steps:

v = {v1, v2, v3}
w = {w1, w2, w3}
q = Simplify[Norm[v]^2,
p = Simplify[Norm[w]^2,
x = Simplify[Dot[v, w]^2]
a = Simplify[Expand[q*p - x]]
z = Cross[v, w]
y = Norm[z]^2
y == a

However in computing the Norms of the vector, it gives me Abs[v1], Abs[v2], Abs[v3], etc. Any idea on how to remove it? Or should I define the elements of the vectors to be real numbers?

Answer 1

Defining the vectors:

v1 = Array[v, 3]
v2 = Array[w, 3]
(* {v[1], v[2], v[3]} *)
(* {w[1], w[2], w[3]} *)

Then:

Cross[v1, v2].Cross[v1, v2] - (v1.v1 v2.v2 - (v1.v2)^2) // Expand
(* 0 *)

Note that in general, when doing algebraic manipulations, it is better to use the explicit form of the squared-norm of a vector, because Norm is interpreted in terms of Abs which does not automatically simplify because Mathematica assumes all symbols are Complex unless told otherwise. In other words, use Sqrt[v.v] instead of Norm[v] and v.v instead of Norm[v]^2.