Quantities without unit of measure

[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...

 

[haruspex]

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.

See if any of these are useful:
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-forces/
https://www.physicsforums.com/insights/frequently-made-errors-pseudo-resultant-forces/
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-moments/
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-momentum-impacts/

 

[Caio Graco]

See if any of these are useful:
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-forces/
https://www.physicsforums.com/insights/frequently-made-errors-pseudo-resultant-forces/
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-moments/
https://www.physicsforums.com/insights/frequently-made-errors-mechanics-momentum-impacts/

Ok. Thanks.