# Scalar Product of Orthonormal Basis: Equal to 1?

• Tony Stark
In summary, 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.
Tony Stark
What is the scalar product of orthonormal basis? is it equal to 1
why is a.b=ηαβaαbβ having dissimilar value

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.

Tony Stark
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.

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.

## What is the definition of scalar product of orthonormal basis?

The scalar product of orthonormal basis is a mathematical concept that involves multiplying two vectors to obtain a scalar value. It is also known as the dot product and is denoted by a dot (·) between the two vectors.

## How is the scalar product of orthonormal basis calculated?

The scalar product of orthonormal basis is calculated by multiplying the corresponding components of the two vectors and then adding the products together. For example, if the two vectors are a = (a1, a2, a3) and b = (b1, b2, b3), then their scalar product would be a1b1 + a2b2 + a3b3.

## Why is the scalar product of orthonormal basis important in mathematics?

The scalar product of orthonormal basis is important in mathematics because it allows us to determine the angle between two vectors, find the projection of one vector onto another, and calculate the magnitude of a vector.

## What is the significance of the scalar product of orthonormal basis equal to 1?

When the scalar product of orthonormal basis is equal to 1, it means that the two vectors are perpendicular to each other. This is because the cosine of 90 degrees (the angle between two perpendicular vectors) is equal to 0, and the scalar product is equal to the product of the magnitudes of the two vectors multiplied by the cosine of the angle between them.

## How is the scalar product of orthonormal basis related to linear independence?

The scalar product of orthonormal basis is closely related to linear independence. If the scalar product of two vectors is equal to 0, it means that the vectors are orthogonal (perpendicular). This indicates that the two vectors are linearly independent, as they are not parallel to each other and cannot be expressed as a multiple of one another.

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