Defining Right-Handed Coordinate Systems in Non-Orthogonal Bases

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Discussion Overview

The discussion revolves around defining right-handed coordinate systems in non-orthonormal bases, exploring the criteria and implications of such definitions. Participants examine the application of the right-hand rule, the scalar triple product, and the determinant of basis vectors in determining handedness.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to define a right-handed coordinate system for a non-orthonormal basis.
  • Another participant suggests that using the right-hand rule often leads to non-orthonormal bases, particularly when the angles between vectors are oblique.
  • It is noted that any three vectors that are not pairwise collinear and form a right-hand triad can be used.
  • One participant mentions that the scalar triple product is used to determine handedness, with positive values indicating right-handed systems and negative values indicating left-handed systems.
  • Another participant points out that the definition of the vector product is crucial, as it inherently includes the concept of right-handedness.
  • It is proposed that a positive determinant of the basis vectors in their natural order indicates a right-handed system, while a negative determinant indicates a left-handed system, with zero indicating an error in the setup.

Areas of Agreement / Disagreement

Participants express various viewpoints on the definition and implications of right-handedness in non-orthonormal bases, with no consensus reached on a singular definition or method.

Contextual Notes

Participants highlight the importance of the definitions used for vector products and determinants, suggesting that these may influence the interpretation of handedness in non-orthonormal contexts.

1MileCrash
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How do you define a right handed coordinate system for a basis which is not orthonormal?
 
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When you use the right hand rule and actually try to do that with your hand, you almost always get a non-orthonormal basis. If the angles between you fingers are very oblique, that is a good example; non of the angles may be zero, however.

More formally, it should be any three vectors not pairwise collinear, and forming a right-hand triad.
 
Book just says scalar triple product positive for right hand, negative for left...
 
Yeah, it boils down to that. One caveat is that it depends on the definition of the vector product, which itself includes right-handedness.
 
That's interesting.. so really, if it is negative it means it "differs" from the way we have defined the vector product (right handed.)

Thanks.
 
It basically boils down to a positive determinant of the all the basis vectors in their natural order. If it's positive then it's right handed, if it's negative it's left handed (you can't get zero and if you do it means you've done something wrong).
 

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