# Quantities without unit of measure

## Homework Statement

Quantities without unit of measure are:
A) Necessarily scalar
B) Necessarily vector
C) Can be scalar or vector
D) They are neither scalar nor vector

No equations

## The Attempt at a Solution

For example, the refractive index of a medium is a number without unit of measure. If it were to risk between climbing or vector, I'd say climb. However, anywhere scale is defined by a number accompanied by its unit. Therefore, the index of refraction would not be scalar and since it has no orientation, it is also not a vector, so what would it be?

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berkeman
Mentor

## Homework Statement

Quantities without unit of measure are:
A) Necessarily scalar
B) Necessarily vector
C) Can be scalar or vector
D) They are neither scalar nor vector

No equations

## The Attempt at a Solution

For example, the refractive index of a medium is a number without unit of measure. If it were to risk between climbing or vector, I'd say climb. However, anywhere scale is defined by a number accompanied by its unit. Therefore, the index of refraction would not be scalar and since it has no orientation, it is also not a vector, so what would it be?
The simple refractive index of an isotropic dielectric is a scalar, correct. But what if the dielectric is not isotropic...?

The simple refractive index of an isotropic dielectric is a scalar, correct. But what if the dielectric is not isotropic...?
Is it even that the refractive index of an isotropic dielectric is a scalar? We are sure that this is a pure number, that is, without a unit of measure. But does not having unity still characterize it as scalar?
And in case if the dielectric is not isotropic, there will be a tensor that will represent the index of refraction.

berkeman
Mentor
Is it even that the refractive index of an isotropic dielectric is a scalar? We are sure that this is a pure number, that is, without a unit of measure. But does not having unity still characterize it as scalar?
It is certainly a scalar (no direction) in the vacuum, and also in isotropic media. https://en.wikipedia.org/wiki/Relative_permittivity#Terminology
And in case if the dielectric is not isotropic, there will be a tensor that will represent the index of refraction.
Very good! So I think you have what you need to answer the question now (a good example).

It is certainly a scalar (no direction) in the vacuum, and also in isotropic media. https://en.wikipedia.org/wiki/Relative_permittivity#Terminology

Very good! So I think you have what you need to answer the question now (a good example).
So to close, in case you were to mark my initial question, you would mark the letter A. Is that it?

Quantities without unit of measure are:
A) Necessarily scalar
B) Necessarily vector
C) Can be scalar or vector
D) They are neither scalar nor vector

haruspex
Homework Helper
Gold Member
Is a tensor an example of a vector?
I think it's the reverse: a vector is an example of a tensor.

haruspex
Homework Helper
Gold Member
I think it's the reverse: a vector is an example of a tensor.
Sure, but that does not make a dimensionless tensor an example of a dimensionless vector.

Yeah, looking through the Wikipedia list doesn't seem to show vectors....
Does the wiki show any scalar list that shows refractive index as an example? I do not think so.

haruspex
Homework Helper
Gold Member
Yeah, looking through the Wikipedia list doesn't seem to show vectors.... https://en.wikipedia.org/wiki/List_of_dimensionless_quantities
Seems to me that dimensionless constants associated with "the universe" are necessarily scalars by the principle of isotropy.
For attributes of materials, they will surely be scalars or tensors. (And in the scalar case, arguably just a degenerate tensor.)
Can't think what else...

Sure, but that does not make a dimensionless tensor an example of a dimensionless vector.
A dimensionless vector must be an example of a non-dimensional tensor. I believe.

haruspex
Homework Helper
Gold Member
A dimensionless vector must be an example of a non-dimensional tensor. I believe.
Yes, but that does not help you. You have found a dimensionless tensor, and the example you need is of a dimensionless vector. So you would need to argue that a tensor is an example of a vector.

Yes, but that does not help you. You have found a dimensionless tensor, and the example you need is of a dimensionless vector. So you would need to argue that a tensor is an example of a vector.
What letter would you mark for the initial problem? Please...

Quantities without unit of measure are:
A) Necessarily scalar
B) Necessarily vector
C) Can be scalar or vector
D) They are neither scalar nor vector

Can I say that a normal one of a surface which I wish to calculate, for example, the electric flow is an example of dimensionless vector quantity?

I'm sorry for my english.

berkeman
Mentor
Can I say that a normal one of a surface which I wish to calculate, for example, the electric flow is an example of dimensionless vector quantity?
Electric flow would have dimensions (like Coulombs per second), but you bring up an interesting point. Unit vectors by definition do not have units... I don't know if they qualify as dimensionless "quantites", though. https://en.wikipedia.org/wiki/Unit_vector
I'm sorry for my english.
Not a problem. It's much better than any of my own foreign language skills.

fresh_42
Mentor
A single quantity without unit is only a number, that is another word for a scalar. But one can always regard several of them and arrange them as a vector of scalars. The question then is, whether such an arrangement can be considered to be a quantity or not. The refractive index is a good example as well as a non isotropic medium is. One could as well consider ##(r,s,t) = \textrm{ refractive index air to (oil, water, glas) }## as a single quantity instead. The answer is thus a matter of definition, so it's probably more of a question what the teacher wants to hear rather than a question with a unique answer, except for "quantity" is defined somewhere.

berkeman
Electric flow would have dimensions (like Coulombs per second), but you bring up an interesting point. Unit vectors by definition do not have units... I don't know if they qualify as dimensionless "quantites", though.
I think you misunderstood me. I did not refer to the unit of electric flow, but to the fact that normal has no unit of measure, to be a unit vector. And the question is whether normal would be an example of dimensionless vector quantity.

Not a problem. It's much better than any of my own foreign language skills.
Not a problem. It's much better than any of my own foreign language skills.
My English is all coming from google translator. I know nothing of English, only Brazilian Portuguese hehehe.

berkeman
A single quantity without unit is only a number, that is another word for a scalar.
See the first paragraph of this pdf and notice that it is said that to be a scalar magnitude is necessary to have unit of measurement.
Is it correct?

http://zeus.plmsc.psu.edu/~manias/MatSE447/03_Tensors.pdf

The answer is thus a matter of definition, so it's probably more of a question what the teacher wants to hear rather than a question with a unique answer, except for "quantity" is defined somewhere.
This question was elaborated by me!

fresh_42
Mentor
See the first paragraph of this pdf and notice that it is said that to be a scalar magnitude is necessary to have unit of measurement.
Is it correct?

http://zeus.plmsc.psu.edu/~manias/MatSE447/03_Tensors.pdf
O.k. But as in the example of a refractive index, a quantity can also be a number without a unit. Here you measure a quotient and the units cancel out. This means, unit or not isn't the question to distinguish between scalars and vectors. Nor is it a matter of dimension. Mathematically a single number (scalar) can also be a one-dimensional vector. Physically a dimensionless scalar, is a number as the refraction index is. The hydrostatic pressure has a physical dimension (pressure) and temperature is also a physical dimension.

What is meant in the paper is the following:
Say we have the hydrostatic pressure ##H(\vec{x}_0)## and the temperature ##T(\vec{x}_0)## at a point ##\vec{x}_0## on earth.
• ##H_0=H(\vec{x}_0)## as well as ##T_0=T(\vec{x}_0)## are scalars, because they don't have a direction, only a scale - a number. They are not dimensionless and both have a unit, although they are scalars.##\\##
• ##(H_0,T_0) = (H(\vec{x}_0),T(\vec{x}_0))## is a vector of scalars with two different dimensions.##\\##
• ##\{\,(\,H(\vec{x})\,,\, \vec{x}\,)\,\vert \, \vec{x} \textrm{ a point on earth }\}## is a scalar field: each point ##\vec{x}## has an associated scalar ##H(\vec{x})## which gives us a field of scalars. Usually the entire scalar field is also briefly noted as ##H## or ##H(\vec{x})##.##\\##
• ##\{\,(\,(H(\vec{x}),T(\vec{x}))\,,\, \vec{x}\,)\,\vert \, \vec{x} \textrm{ a point on earth }\}## is a vector field: each point ##\vec{x}## has an associated vector ##(H(\vec{x}),T(\vec{x}))## which gives us a field of vectors; each vector being a pair of scalars. Usually the entire vector field is either written as ##(H,T)=(H(\vec{x}),T(\vec{x}))## or abbreviated with another letter ##V##.##\\##
• If we define ##\vec{w}(\vec{x}_0)## as the vector of wind at a point ##\vec{x_0}## on earth, then ##\vec{w}## is the classical vector with a direction, where the wind blows, and a magnitude, how strong the wind blows.##\\##
• If we define ##\{\,(\,\vec{w}(\vec{x}),\vec{x}\,)\,\vert \,\vec{x} \textrm{ a point on earth }\}## then we get a classical vector field, that is to each point ##\vec{x}## an associated vector of wind ##\vec{w}(\vec{x})##. Usually this construction is only denoted by, e.g. ##W## or ##W(\vec{x})\,.## ##\\##