Scalar Product of Orthonormal Basis: Equal to 1?

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Discussion Overview

The discussion revolves around the scalar product of vectors in an orthonormal basis, specifically whether it is equal to 1 and the implications of the terms used in this context. Participants explore definitions and interpretations related to scalar products, inner products, and the properties of orthonormal bases.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the concept of a scalar product of a basis, suggesting the question pertains to the scalar product between two vectors in an orthonormal basis.
  • Another participant states that the inner product of two different basis vectors is zero, while the inner product of a basis vector with itself yields ±1, depending on the nature of the vector (timelike or spacelike).
  • There is a clarification that the term "inner product" is synonymous with "dot product" and "scalar product," although some participants express reservations about the clarity of related definitions.
  • One participant corrects another regarding terminology, emphasizing that the term "ortho" refers to orthogonality (zero product for different vectors) and "normal" refers to the self-product yielding ±1.
  • A participant suggests that the original question could be interpreted in various ways and expresses a desire to clarify the intent of the original poster.
  • There is a mention of a specific text (Hartle) that may cover the topic, although the participant does not have access to it.

Areas of Agreement / Disagreement

Participants generally agree on the properties of the scalar product in an orthonormal basis, but there is some contention regarding terminology and interpretation of the original question. No consensus is reached on the exact phrasing or implications of the terms used.

Contextual Notes

Some assumptions about the definitions of scalar products, inner products, and the nature of basis vectors remain unresolved. The discussion also reflects varying interpretations of the original question posed by the first participant.

Tony Stark
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What is the scalar product of orthonormal basis? is it equal to 1
why is a.b=ηαβaαbβ having dissimilar value
 
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There is no such thing as a scalar product of a basis. The only possible interpretation of your question is what the scalar product between two vectors in an orthonormal basis is. The answer is that if you take two different vectors of the basis, it is zero (this is the "normal" part of orthonormal) while if you take the inner product of one of the vectors in the basis with itself you get ±1 depending on whether you chose a timelike or spacelike basis vector.
 
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Orodruin said:
There is no such thing as a scalar product of a basis. The only possible interpretation of your question is what the scalar product between two vectors in an orthonormal basis is. The answer is that if you take two different vectors of the basis, it is zero (this is the "normal" part of orthonormal) while if you take the inner product of one of the vectors in the basis with itself you get ±1 depending on whether you chose a timelike or spacelike basis vector.
What do you mean by INNER PRODUCT
 
Orodruin said:
The answer is that if you take two different vectors of the basis, it is zero (this is the "normal" part of orthonormal)

Actually, it's the "ortho" part, correct? The "normal" part is the ##\pm 1## you get when you take the inner product of a basis vector with itself.
 
PeterDonis said:
Actually, it's the "ortho" part, correct? The "normal" part is the ##\pm 1## you get when you take the inner product of a basis vector with itself.

Indeed, writing faster than thinking. :rolleyes:
 
Scalar product, dot product, and inner product all mean the same thing AFAIK - wiki agrees, though I wasn't very fond of the rest of the wiki article, entitled "dot product".

I'd also interpret
What is the scalar product of orthonormal basis?

as "what is the scalar (or dot, or inner) product of the basis vectors in an orthonormal basis", though I'd stop short of insisting that that's what the OP must have meant in favor of attempting to try and find out if that's what they meant. I agree with the answers that have already been given - I hope they have cleared things up for the OP.

I also can't resist asking - doesn't Hartle cover this? I don't have the book, I would have thought it would have been covered in the text.
 

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