How Do Rotation Matrices Impact Coordinate Systems and Object Transformations?

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Discussion Overview

The discussion revolves around the geometric and intuitive understanding of rotation matrices, particularly in the context of transforming between two sets of coordinates before and after a rotation. Participants explore the implications of these transformations on coordinate systems and objects.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants seek geometric and intuitive insights into rotation matrices and their impact on coordinate systems.
  • There is a discussion about the distinction between rotating the axes versus rotating the vector itself, with some participants noting the difficulty in distinguishing these concepts.
  • One participant suggests that it is beneficial to write calculations step by step to avoid confusion and errors, emphasizing the importance of clarity in understanding transformations.
  • Another participant expresses a differing view on the speed of thinking versus writing, suggesting that writing down steps can help focus cognitive resources on solving specific problems.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of distinguishing between changing coordinates and changing objects in the context of rotation matrices. However, there are differing opinions on the best approach to manage this complexity, particularly regarding the cognitive processes involved in writing versus thinking.

Contextual Notes

Some assumptions about the understanding of rotation matrices and their applications may be missing, as well as potential dependencies on definitions of coordinate systems and transformations.

Leo Authersh
Can anyone give me geometric and intuitive insight on Rotation matrices which has two sets of coordinates after Transformation?
 

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fresh_42 said:
Can you be a bit more precise? What don't you understand exactly? And what, which cannot be found already here:
https://en.wikipedia.org/wiki/Rotation_matrix
Hi, Now I understood. In the above picture, they have the vector fixed and rotated the axes. I couldn't understood it at first. Thank you for your answer :)
 
I think the post refers to two coordinate systems. Before- and After- rotation.
 
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WWGD said:
I think the post refers to two coordinate systems. Before- and After- rotation.
Yes it is. Here they rotated the axes instead of vector.
 
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Leo Authersh said:
Yes it is. Here they rotated the axes instead of vector.
Yeah, this is really a bit difficult to distinguish sometimes: changing the coordinates or changing the object and how does it affect the matrix? It's easy to get confused if one doesn't keep track of what is what and in which coordinate system. A tip is better to write some extra lines than to search for errors afterwards. I always tell students when it comes to calculations: write it down step by step, because writing is fast, thinking is slow.
 
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fresh_42 said:
Yeah, this is really a bit difficult to distinguish sometimes: changing the coordinates or changing the object and how does it affect the matrix? It's easy to get confused if one doesn't keep track of what is what and in which coordinate system. A tip is better to write some extra lines than to search for errors afterwards. I always tell students when it comes to calculations: write it down step by step, because writing is fast, thinking is slow.
I believe thinking is fast. But if we write down each step we can save all the cognition for solving a specific step which otherwise would be wasted in memorizing the already solved steps. But that's just for me :)
 

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