Mathematically, an "angle" by itself *is* a scalar, and I'm not arguing
otherwise. I'm saying that "Angle Of Attack" requires direction to have
meaning. Without direction, there is no AOA.
[...]

Well, OK. Then, how do you determine the AOA when the aircraft is parked?
If the component of direction is inseparable from the definition of AOA,
how can it be a scalar?

How long is it between 11 AM and 1:30 PM? The answer is 150 minutes.
Now, is that 150 minutes AM or 150 minutes PM?

In order to form the difference, you NEED to know whether the original
times are AM or PM, but the result is a pure number of minutes. 11 AM
is a =time of day=, 1:30 pm is a =time of day= but the difference is
=not= a time of day, it is just a number (of minutes).

In a similar vein, in order to form an angle (a pure number), you need
to have not just one direction, but two. You need TWO quantites that
have direction (they don't even have to have magnitude!). However, the
result (the angle between them) has no direction (beyond the algebraic
sign).

When you say:

I'm saying that "Angle Of Attack" requires direction to have
meaning.

all you're really saying is that you don't have an angle of attack if
you don't have the requisite components (a relative wind, and a chord).
But don't confuse the components with the result. Area is made up of
length and width, but area is not in itself one dimesional. Cakes are
made with raw eggs and flour, but I'm not likely to confuse the two any
time soon.

Jose
--
"Never trust anything that can think for itself, if you can't see where
it keeps its brain."
(chapter 10 of book 3 - Harry Potter).
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