How to form the transformation matrix for this

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Discussion Overview

The discussion revolves around forming a transformation matrix that rotates the x1 axis of a rectangular coordinate system by 60 degrees toward the x2 and x3 axes. Participants explore the ambiguity of the term "rotating one axis toward the two other" and the implications for determining angles and the nature of the rotation.

Discussion Character

  • Exploratory, Debate/contested, Conceptual clarification

Main Points Raised

  • One participant expresses confusion about whether to rotate the x1 axis toward the x2-x3 plane or in another manner, questioning how to determine the angles between the rotated x1 axis and the x2 and x3 axes.
  • Another participant points out that there are infinitely many ways to rotate the x1 axis by 60 degrees toward the x2-x3 plane, suggesting various fixed points and directions for the rotation.
  • Some participants speculate that the question might involve separate rotations toward the x2 and x3 axes, or a combined rotation around the x3 axis, leading to different interpretations of the problem.
  • A later reply indicates that clarification from a teacher confirmed the rotation should be "around the x3 axis," which simplifies the problem for one participant.

Areas of Agreement / Disagreement

Participants generally agree on the ambiguity of the original question and the various interpretations of the rotation. However, there is no consensus on the specific method to approach the transformation matrix until the clarification from the teacher is introduced.

Contextual Notes

The discussion highlights limitations in understanding the problem's requirements, particularly regarding the definitions of rotation and the angles involved. The lack of clarity in the original question contributes to the uncertainty expressed by participants.

Byang
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We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis.
The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or does it mean something else? I tried to rotate x1 to the x2-x3 plane but I can never be sure of the angle between x1' and x2, and x1' and x3. How do I find the angles?
 
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Byang said:
We were asked to form the transformation matrix that rotates the x1 axis of a rectangular coordinate system 60 degrees toward x2 and the x3 axis.
The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x1 60 degrees toward the x2-x3 plane or does it mean something else? I tried to rotate x1 to the x2-x3 plane but I can never be sure of the angle between x1' and x2, and x1' and x3. How do I find the angles?

There are infinitely many possible ways of rotating the ##x_1## axis by ##60^o## towards the ##x_2-x_3## plane. For example, you could fix ##x_2 = 0## and rotate the ##x_1## axis towards the ##x_3## axis, or fix ##x_3 = 0## and rotate ##x_1## towards ##x_2##, etc. In all these cases you would be rotating ##x_1## by ##60^o## towards the ##x_2-x_3## plane. In general, you could fix an arbitrary point ##p = (0,p_2,p_3)## in the ##x_2-x_3## plane, and rotate the ##x_1## axis towards ##p##.

Basically, I don't know exactly what your question wants; perhaps they mean ##60^o ## towards ##x_2## and then ##60^o## towards ##x_3##, or maybe they mean two separate questions---one for an ##x_1 \to x_2## rotation and another for an ##x_1 \to x_3## rotation.
 
Ray Vickson said:
Basically, I don't know exactly what your question wants
Haha, that's exactly my problem. But thank you for answering. We've cleared it up with our teacher and she said it was supposed to be "around x3 axis," which makes the problem a lot easier.
 
Thread moved as it seems to be more of a general question than a specific homework problem.
 

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