# Homework Help: Quantities without unit of measure

1. Feb 23, 2017

### Caio Graco

1. The problem statement, all variables and given/known data
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

2. Relevant equations
No equations

3. 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?

2. Feb 23, 2017

### Staff: Mentor

The simple refractive index of an isotropic dielectric is a scalar, correct. But what if the dielectric is not isotropic...?

3. Feb 23, 2017

### Caio Graco

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.

4. Feb 23, 2017

### Staff: Mentor

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).

5. Feb 23, 2017

### Caio Graco

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

6. Feb 23, 2017

### haruspex

7. Feb 23, 2017

### Caio Graco

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

8. Feb 23, 2017

### Staff: Mentor

9. Feb 23, 2017

### haruspex

Sure, but that does not make a dimensionless tensor an example of a dimensionless vector.

10. Feb 23, 2017

### Staff: Mentor

11. Feb 23, 2017

### Caio Graco

Does the wiki show any scalar list that shows refractive index as an example? I do not think so.

12. Feb 23, 2017

### haruspex

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...

13. Feb 23, 2017

### Caio Graco

A dimensionless vector must be an example of a non-dimensional tensor. I believe.

14. Feb 23, 2017

### haruspex

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.

15. Feb 23, 2017

### Caio Graco

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

16. Feb 23, 2017

### Caio Graco

17. Feb 23, 2017

### 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?

18. Feb 23, 2017

### Caio Graco

I'm sorry for my english.

19. Feb 23, 2017

### Staff: Mentor

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
Not a problem. It's much better than any of my own foreign language skills.

20. Feb 23, 2017

### Staff: 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.

21. Feb 23, 2017

### Caio Graco

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.

22. Feb 23, 2017

### Caio Graco

My English is all coming from google translator. I know nothing of English, only Brazilian Portuguese hehehe.

23. Feb 23, 2017

### Caio Graco

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

24. Feb 23, 2017

### Caio Graco

This question was elaborated by me!

25. Feb 23, 2017

### Staff: Mentor

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