Reciprocal basis and orientation

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SUMMARY

The discussion focuses on proving that two reciprocal bases, specifically (e_1, e_2, e_3) and (e^1, e^2, e^3), share the same orientation, either both being right-handed or both left-handed. The key argument is based on the scalar triple product and the relationship between the volumes V and V' of the parallelepipeds formed by these bases. By demonstrating that VV' equals 1, it is established that both bases have the same orientation, confirming their equivalence in terms of handedness.

PREREQUISITES
  • Understanding of vector calculus and scalar triple products
  • Familiarity with the concept of reciprocal bases in linear algebra
  • Knowledge of vector cross products and their properties
  • Basic principles of orientation in three-dimensional space
NEXT STEPS
  • Study the properties of scalar triple products in vector spaces
  • Explore the concept of reciprocal bases in more depth
  • Learn about vector cross products and their applications in geometry
  • Investigate the implications of orientation in higher-dimensional spaces
USEFUL FOR

This discussion is beneficial for mathematicians, physicists, and students studying linear algebra or vector calculus, particularly those interested in the properties of vector spaces and orientation.

feynman137
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How to prove that two reciprocal basis are either both right ended or both left-handed? If (e_1,e_2,e_3) and (e^1,e^2,e^3) are two such basis, since the scalar triple products depend on orientation, it would be enough to show that VV'=1 (where V and V' are the volumes, taken with their sign, of the two parallelepipeds obtained from the two basis vectors). How to do it?
 
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I figured it out: just write (e_2 x e_3).(e^2 x e^3) as ((e_2 x e_3) x e^2).e^3 and by basic identities we can prove that VV'=1>0, so the two basis have the same orientation.
 

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