# How to trasform an orthonormal system in two reference frames

1. Apr 26, 2012

### 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?

2. Apr 26, 2012

### tiny-tim

hi matteo86bo!

i don't understand

x' = x - xc.o.m

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

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