How to trasform an orthonormal system in two reference frames

In summary, the conversation discusses the problem of defining a new orthonormal system in a different reference frame and recovering its components in the original system. The proposed method involves adding the center of mass to the new system, but this results in a non-orthonormal system. The conversation ends with confusion about the method and its correctness.
  • #1
matteo86bo
60
0
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 [itex]\vec x{_{cm}}[/itex] of the system.
Now I move the COM reference frame x'y'z':
[itex]\vec{x'}=\vec{x}-\vec x{_{cm}}[/itex]

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:

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

I don't recover an orthonormal system of axis! What is wrong in my method?
 
Mathematics news on Phys.org
  • #2
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
 

FAQ: How to trasform an orthonormal system in two reference frames

How do you define an orthonormal system?

An orthonormal system is a set of vectors that are mutually perpendicular and have a unit length. This means that the dot product of any two vectors in the system is equal to 0, and the norm (magnitude) of each vector is equal to 1.

Why would you need to transform an orthonormal system in two reference frames?

In certain situations, it may be necessary to work with coordinates or vectors in different reference frames. By transforming an orthonormal system, we can easily convert between the two frames and perform calculations or analyses.

What is the process for transforming an orthonormal system?

The process for transforming an orthonormal system involves multiplying the original system's vectors by a transformation matrix. This matrix is typically composed of the basis vectors of the new reference frame, which are expressed in terms of the old reference frame's basis vectors.

Can an orthonormal system be transformed into any reference frame?

Yes, an orthonormal system can be transformed into any reference frame as long as the basis vectors of the new frame are expressed in terms of the old frame's basis vectors. This allows for easy conversion between different coordinate systems.

What are the benefits of using an orthonormal system in multiple reference frames?

An orthonormal system is a useful tool in various scientific fields, such as physics and engineering, as it allows for accurate and efficient calculations in different reference frames. It also simplifies the process of converting between coordinates and vectors in different frames, making it a valuable tool for modeling and analysis.

Similar threads

Kinetic energy in center of mass reference frame
Replies
3
Views
813
I Principle of relativity and Galileo's group
Replies
10
Views
1K
Representing a function in a different space
Replies
1
Views
2K
Restrictions on the frame of reference
Replies
4
Views
1K
Is angular momentum conserved in a non-inertial frame?
Replies
14
Views
2K
I Reference frame vs coordinate chart
2
Replies
62
Views
5K
Spring-mediated dipole in a magnetic field
Replies
31
Views
2K
Finding Probability of Two-Identical-Particle System in a Given State
Replies
16
Views
2K
I Different reference frames in QM
Replies
10
Views
1K
Calculate center of rotation of monocopter
Replies
17
Views
1K

Hot Threads

Recent Insights

Back
Top