it is valid for a vector to have a magnitude of zero.

Correct.

It is NOT
valid for a scalar to have a directional component

Correct.

and it is meaningless
to have an AOA with no directional component and magnitude

Incorrect. I can give you many examples of such AOAs. Can you give me
an example of an AOA that =itself= has a direction and magnitude? (Not
that it's derived from things that have direction and magnitude, but
that it, =itself= has such)

The two aspects of the AOA is referenced to the wing chord and relative
wind, not the fuselage.

The two aspects of the AOA are referenced to each other. I refereneced
them to the same other thing (fuselage) and then derived their relation
to each other.

I'd say that it is often "OK" to PRESUME the directional components and
IGNORE their value if they are unimportant to usages where only the angle
is needed.

It is not OK to presume anything in math. Things are what they are
defined to be.

You might be thinking of "unit vectors" in which case a magnitude of one
is used, but they are defined that way. Or you might be thinking of the
algebraic sign (which is part of a scalar quantity).

We're not talking about generic "angles", but an "Angle Of Attack"

An angle of attack =is= an angle. All angles are scalars. Therefore,
an angle of attack is a scalar. Which part of this do you disagree with?

i.e.,
a specific usage which is defined by and inseparable from the components
of motion (aka relative wind).

Defined by, yes. Inseperable from, no.

The price to earnings ratio (PE) of a stock is =defined by= the dollar
price of a stock, and the dollar earnings of the company divided by the
number of shares outstanding. Without those components, you don't have
a PE ratio. But the PE is a pure number. It is not a dollar amount.

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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