How to trasform an orthonormal system in two reference frames

[matteo86bo]

My question is not homework. I feel ashamed of having this doubts but I'm really stuck on this.
The problem is I have a reference frame xyz and here I define the COM \vec x{_{cm}} of the system.
Now I move the COM reference frame x'y'z':
\vec{x'}=\vec{x}-\vec x{_{cm}}

In this reference frame I define a new orthonormal system x''y''z'' centered in (0,0,0), i.e. the COM mass.

I now want to recover to component of my last orthonormal system x''y''z'' in the original system xyz.

If I do:

\vec{x''}{ {\rm (in~ xyz)}}=\vec{x''}{ {\rm (in~ x'y'z')}}+\vec x{_{cm}}

I don't recover an orthonormal system of axis! What is wrong in my method?

 

[tiny-tim]

hi matteo86bo!

i don't understand

x' = x - x_c.o.m

so your xyz directions are the same, and only the origin has changed

then x'' = x' + x_c.o.m = x