Reciprocal basis and orientation

In summary, proving that two reciprocal basis are either both right ended or both left-handed can be done by showing that VV'=1, where V and V' are the volumes of the two parallelepipeds obtained from the two basis vectors. This can be achieved by writing (e_2 x e_3).(e^2 x e^3) as ((e_2 x e_3) x e^2).e^3 and using basic identities to prove that VV'=1>0, indicating that the two basis have the same orientation.
  • #1
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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  • #2
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.
 

1. What is a reciprocal basis?

A reciprocal basis is a set of vectors that are perpendicular to the original basis vectors in a vector space. The dot product of a basis vector and its corresponding reciprocal basis vector is equal to 1, and the dot product of any two reciprocal basis vectors is equal to 0.

2. How is a reciprocal basis calculated?

A reciprocal basis can be calculated by taking the cross product of any two basis vectors and normalizing the resulting vector. This process is repeated for each basis vector to obtain a set of reciprocal basis vectors.

3. What is the significance of a reciprocal basis?

A reciprocal basis is important in understanding the orientation of a vector space. It helps to define the direction and magnitude of vectors in a space and is essential in many mathematical and physical applications, such as crystallography and Fourier analysis.

4. How does the orientation of a reciprocal basis relate to the orientation of the original basis?

The orientation of a reciprocal basis is perpendicular to the orientation of the original basis. This means that if the original basis vectors are aligned in a specific direction, the reciprocal basis vectors will be aligned in a direction perpendicular to it.

5. Can a vector space have more than one reciprocal basis?

Yes, a vector space can have an infinite number of reciprocal bases. This is because the cross product of any two basis vectors can be multiplied by a scalar value, resulting in a different but still valid reciprocal basis. However, all reciprocal bases for a given vector space will have the same orientation and properties.

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