Quantities without unit of measure

[Caio Graco]

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

Justify your answer!

Homework Equations

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?

 

[berkeman]

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

Justify your answer!

Homework Equations

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...?

 

[Caio Graco]

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]

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).

 

[Caio Graco]

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]

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).

Is a tensor an example of a vector?

 

[Caio Graco]

Is a tensor an example of a vector?

I think it's the reverse: a vector is an example of a tensor.

 

[berkeman]

Is a tensor an example of a vector?

Hmm, good question. Maybe I was too hasty:

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

 

[haruspex]

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.

 

[berkeman]

Yeah, looking through the Wikipedia list doesn't seem to show vectors... https://en.wikipedia.org/wiki/List_of_dimensionless_quantities

 

[Caio Graco]

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]

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...

 

[Caio Graco]

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]

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.

 

[Caio Graco]

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

 

[Caio Graco]

Yeah, looking through the Wikipedia list doesn't seem to show vectors... https://en.wikipedia.org/wiki/List_of_dimensionless_quantities

Are there scalar, vector, and dimensionless quantities?
What do you think?

 

[Caio Graco]

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?

 

[Caio Graco]

I'm sorry for my english.

 

[berkeman]

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]

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.

 

[Caio Graco]

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.

 

[Caio Graco]

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.

 

[Caio Graco]

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

 

[Caio Graco]

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]

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 $W(\vec{x})\,.$ $\\$

 

[haruspex]

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

I would not take that web page as an authoritative source on such fine distinctions. It seems fairly low level.
A good example of a naturally occurring dimensionless scalar is the Reynolds number. Another is π.

 

[Caio Graco]

I would not take that web page as an authoritative source on such fine distinctions. It seems fairly low level.
A good example of a naturally occurring dimensionless scalar is the Reynolds number. Another is π.

Do you know of any reliable sources on the WEB that cite the examples you mentioned, explicitly called them scalar quantities?

 

[Caio Graco]

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 $W(\vec{x})\,.$ $\\$

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

 

[fresh_42]

Do you know of any reliable sources on the WEB that cite the examples you mentioned, explicitly called them scalar quantities?

Why do you insist on this wording? It's important to understand what is meant by scalar, vector, quantity, unit and dimension. Among these, quantity is certainly the least rigorously defined term.

 

[fresh_42]

Quantities can be a scalar or a vector (C). And it doesn't matter whether they have a unit or not, only that they can be measured somehow. This makes them different from a quality.

 

[Caio Graco]

Why do you insist on this wording? It's important to understand what is meant by scalar, vector, quantity, unit and dimension. Among these, quantity is certainly the least rigorously defined term.

I am very curious to know how to resolve this problem in detail, as I have never seen this discussed in any book. I am a physics teacher and I am writing a book on vectors in which I wish to insert such a discussion.

 

[Caio Graco]

Quantities can be a scalar or a vector (C). And it doesn't matter whether they have a unit or not, only that they can be measured somehow. This makes them different from a quality.

I apologize for the persistence and thank you for your attention

 

[fresh_42]

I apologize for the persistence and thank you for your attention

De nada.

 

[haruspex]

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

I would consider this a poor question because it is not clear what is allowed, though maybe there is an interesting question hiding within it.
As has been pointed out, it is always possible to construct a vector from a list of arbitrary scalars. If that is permitted then clearly the answer is C.
Also, many dimensionless scalars are so by virtue of being a ratio of two scalar entities of the same dimension. In the same way, you might argue that a unit vector derived from a given (dimensional) vector is a dimensionless vector. The direction of a velocity is given by $\vec v/|\vec v|$. But again, this feels artificial.
If we exclude those then it gets more interesting. As I wrote, there are reasons why dimensionless constants in the real world tend to be either scalars or tensors.

 

[berkeman]

I am a physics teacher and I am writing a book on vectors in which I wish to insert such a discussion.

Interesting twist to this thread...