How to trasform an orthonormal system in two reference frames

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SUMMARY

The discussion focuses on transforming an orthonormal system between two reference frames, specifically from xyz to x'y'z' and then to x''y''z''. The user defines the center of mass (COM) as \(\vec{x}_{cm}\) and attempts to recover the components of the orthonormal system x''y''z'' in the original xyz frame. The key issue identified is the misunderstanding of how the transformation affects the orthonormality of the axes, as simply translating the origin does not maintain the orthonormal properties of the system.

PREREQUISITES
  • Understanding of orthonormal systems in three-dimensional space
  • Familiarity with coordinate transformations
  • Knowledge of center of mass (COM) calculations
  • Basic vector operations in physics
NEXT STEPS
  • Study the principles of coordinate transformations in physics
  • Learn about maintaining orthonormality during transformations
  • Explore vector addition and its implications in different reference frames
  • Investigate the mathematical properties of orthonormal bases
USEFUL FOR

This discussion is beneficial for physics students, researchers in mechanics, and anyone involved in coordinate transformations and vector analysis in three-dimensional space.

matteo86bo
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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?
 
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hi matteo86bo! :smile:

i don't understand :confused:

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
 

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