Why is the special orthogonal group considered the rotation group?

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Discussion Overview

The discussion centers on the nature of the special orthogonal group (SO(3)) and its characterization as the rotation group. Participants explore the conditions that define this group, particularly focusing on the implications of having a determinant of 1 for rotation matrices and the preservation of orientation in rotations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants explain that the special orthogonal group consists of matrices that satisfy the conditions x·x=I and det(x)=1, where I is the identity matrix.
  • One participant argues that a pure rotation should not change the volume, implying that the determinant of a rotation operator must be 1, as it represents the scale factor for volume elements.
  • Another participant questions why the determinant cannot be -1, suggesting that rotations preserve the orientation of a basis, specifically noting that right-handed bases remain right-handed under rotation.
  • One participant proposes that to prove rotations have a determinant of 1, one must first define what constitutes a rotation, suggesting definitions based on group membership in SO(3) or O(3).
  • There is a clarification regarding the notation used, with one participant correcting the expression to include the transpose, indicating a common point of confusion.

Areas of Agreement / Disagreement

Participants express varying degrees of understanding about the determinant condition, with some agreeing on the necessity of a determinant of 1 for proper rotations, while others highlight the need for clearer definitions and reasoning. The discussion remains unresolved regarding the implications of different definitions of rotation.

Contextual Notes

Participants note the importance of definitions in the discussion, particularly regarding the terms "rotation" and "proper rotation," which may influence the understanding of the determinant condition.

tensor33
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I understand that the special orthogonal group consists of matrices x such that x\cdot x=I and detx=1 where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule x\cdot x=I are matrices involved with rotations because they preserve the dot products of vectors. The part I don't get is why the matrices involved with rotation must have determinant 1.
 
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A pure rotation should not dilate or shrink any volume. The determinant of a linear operator is the scale factor by which the volume element increases. Hence, the determinant of a rotation operator must be 1.
 
tensor33 said:
I understand that the special orthogonal group consists of matrices x such that x\cdot x=I and detx=1 where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule x\cdot x=I are matrices involved with rotations because they preserve the dot products of vectors. The part I don't get is why the matrices involved with rotation must have determinant 1.

Did you mean x\cdot x^T=I??

Muphrid said:
A pure rotation should not dilate or shrink any volume. The determinant of a linear operator is the scale factor by which the volume element increases. Hence, the determinant of a rotation operator must be 1.

That doesn't really explain why the determinant can't be -1, which is what the OP is asking.
The reason is that rotations preserves the orientation of a basis. If we have a right-handed basis, then rotations of this will be right-handed as well.
As an example of an orthogonal matrix that does not preserve the orientation, you can probably take a reflection.
 
tensor33 said:
I understand that the special orthogonal group consists of matrices x such that x\cdot x=I and detx=1 where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule x\cdot x=I are matrices involved with rotations because they preserve the dot products of vectors. The part I don't get is why the matrices involved with rotation must have determinant 1.
To prove that rotations have determinant 1, you must first define the term "rotation". It's perfectly OK to just take "R is said to be a rotation if R is a member of SO(3)" as the definition, but then there's nothing to prove. Another option is "R is said to be a rotation if R is a member of the largest connected subgroup of O(3)". Then your task would be to prove that the largest connected subgroup is SO(3).

(I actually prefer the terminology that calls members of O(3) rotations, and members of SO(3) proper rotations).
 
micromass said:
Did you mean x\cdot x^T=I??

Ya, I forgot the transpose part.

Thanks to all those who replied, I'm pretty sure I get it now.
 

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