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Old June 17th 05, 04:40 PM
Jose
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I'm not sure that I follow your analogies, here, Jose.

The analogy is merely that you can use one kind of quantity to derive
another kind of quantity. You can use eggs to derive cake, you can use
"time of day" to derive "time", you can use length to derive area, and
you can use vectors to derive scalars.

The simplest example, I suppose, is a ratio. Fifteen kilograms is THREE
times as much as five kilograms. Fifteen inches is THREE times as much
as five inches. The "three" in both cases is the same - it is a pure
scalar quantity. It is the same "three" as the number of fingers on my
hand that are surrounded by other fingers and the number of days in a
long weekend.

Fifteen kilograms is =not= three times as much as five inches. The
units are important when =deriving= the result, but once the result is
correctly derived, it has its own units (or lack of them).

Similarly, two vectors can intersect at an angle. The angle is not a
vector, it is a scalar. As an aside, two vectors (of the same units)
can also define an area; that area is not a vector, it is a scalar (with
units of square fubars, where "fubars" are the unit both vectors are
measured in).

A vector has magnitude and direction. AOA has no direction in and of
itself. To see this, imagine a wing chord which is inclined three
degrees (the leading edge higher) from some reference plane (say, the
fuselage), and a relative wind which is blowing up from ahead and
underneath at an angle of eighteen degrees to that same fuselage, at
seventy knots. This is typical of an approach in a light aircraft.

What is the angle of attack? To be a scalar, it would have just
magnitude (which could include an algebraic sign). To be a vector, it
would have to have magnitude AND direction.

In this case, the angle of attack is twenty-one degrees. It is the
difference between the two angles given (with reference to the same
fuselage). There is no "direction" to this angle (except perhaps an
algebraic sign). So it is not a vector.

One source of confusion arises because in other contexts angles are also
used to define direction, for example wind velocity is a vector whose
angle is a direction component, not a magnigude component. For example,
"zero three zero at ten knots" is a vector, where the magnitude part is
ten knots, and the direction part is 30 degrees East of North. However,
if you put a weight on an old fashioned butcher scale, the pointer moves
through some angle. That angle does =not= represent a direction, it is
a magnitude only, and thus a scalar (related to the weight of the meat
put in the pan). And if you weigh two cuts of meat, note the angles of
the pointer for each weighing, and subtract those angles, the result is
also an angle - a magnitude with no direction component. This is a scalar.

So, depending on context, angles can be magnitudes =or= directions, but
not both at once.

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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