Altitude - Why is it a Scalar?

[ELLE_AW]

Homework Statement

How come altitude of a mountain is a scalar?

Homework Equations

Scalars = only magnitude
Vectors = have magnitude & direction

The Attempt at a Solution

- Doesn't altitude of a mountain have both magnitude and direction (direction being measured straight up 90 degrees to the ground?

 

[Orodruin]

Altitude is a single number that gives the height above sea level. It cannot, for example, point to the north-west.

 

[FactChecker]

Consider how a negative altitude would be handled. Is that the same as a magnitude and direction? Some people would treat it that way and other people would not.

 

[haruspex]

Scalars = only magnitude

That's the one I would challenge.
It is reasonable to define magnitude (of anything to which the term is fairly applied) to be non-negative, but it is not ok to say a scalar cannot be negative.
Though there is an obvious mapping between a field and a one-dimensional vector space over the field, that does not make them the "same". A field has a defined product operator, $\times:\mathcal{F\times F\rightarrow F}$, whereas the vectors of a vector space do not in general have a product operator $\times:\mathcal{V\times V\rightarrow V}$.
Thus, to be able to multiply signed numbers it is necessary to allow that they be neither vectors nor mere magnitudes.

In the specific case of altitudes, multiplying them does not make much sense. So perhaps it is more logical to regard altitude as a one dimensional vector.