How to define a fixed length vector in Mathematica?

Question

I would like to define a 3-D vector as say v={x,y,z}, but I want it to have a fixed norm, say a unit norm |v|=1. How do I impose this condition on v.

Answer 1

To summarize my comments to the question:

In order to get a normalized version of your (real) vector, just use Normalize:

v=Normalize[{1,2,3}]

Note that you can also use other norms in Normalize, see the documentation for that. I shall assume you want the Euclidean one.

Assume we consider a general normalized vector

v= Normalize[{x, y, z}]

In order to get the desired result from Simplify of v.v, you need to tell Mathematica that {x,y,z} are real:

Simplify[v.v, Element[{x, y, z}, Reals]]

1

as Mathematica assumes by default that $x,y,z$, respectively are complex (and obviously, for e.g. a complex $x=i$, we have $Abs[x]^2=Abs[i]^2=1^2\neq-1=i^2=x^2$, so it cannot simplify in general). Telling Mathematic that we use reals, we have $x^2=Abs[x]^2$ and get the desired result.