I just figured out something about vectors. It has been bugging me for
many years, just below the level of conscious perception, and it finally
oozed to the surface.
The word "vector" is commonl used to mean two different things. They are
not unrelated, but they are definitely not the same.
1) In linear algebra class, one writes expressions of the form
where on the RHS the first factor is called a matrix and the second
factor is called a "vector".
To avoid confusion later, I will call this type of vector a /BLoC/,
which is an acronym for Big List of Components.
The LHS of equation (1) is also a BLoC.
You don't need to know anything about the physical significance of the
BLoC to carry out the calculation in equation (1). It's just arithmetic.
Calling a BloC a "vector" is consistent with the usage in computer
programming, where a "vector" is just a one-dimensional array, i.e. a
collection of elements, with no requirement that the collection have
any particular geometrical or physical significance.
We can give a BLoC a name such as
[ a ]
B := [ ] (2)
[ b ]
which just defines B; it doesn't tell us anything we didn't already know.
2) In physics, there is another kind of "vector". To avoid ambiguity, I
will call it a /phector/, meaning /physics vector/. This is an object P
that lives in real space, and has real geometric and physical significance.
There is no unique way to expand the phector P in terms of its components.
That is, without some nontrivial additional information, we cannot assign
any useful meaning to an expression of the form
[ a ]
P = [ ] (3)
[ b ]
because we don't know which observer's basis is to be used. It would make
incomparably more sense to write something like
[ a ]
P = [ ] (4)
[ b ] @ Moe
which means
P = a X_M + b Y_M (4')
and/or
[ p ]
P = [ ] (5)
[ q ] @ Joe
which means
P = p X_J + q Y_J (5')
where X_M and Y_M are the basis phectors in Moe's frame, and X_J and Y_J are
the basis phectors in Joe's frame. Note that (4), (4'), (5), and (5') are
simultaneously and equally valid descriptions of the *same* phector P.
The phector P is shown by the black arrow. It does not rotate. It just sits
there, as a real physical object in space. Joe looks at P from one viewpoint,
as shown by the blue reference frame. Moe looks at P from another viewpoint,
as shown by the red reference frame.
I emphasize that P is not changed if/when we switch from one reference frame
to another. The phector neither knows nor cares who, if anyone, is looking
at it.
Even though the number a is not equal to p, and b is not equal to q, the
following equation is in fact an equality:
[ p ] [ a ]
[ ] = [ ] (6)
[ q ] @ Joe [ b ] @ Moe
It's like writing 2*12 = 3*8 ... it's two expressions for the same thing.
In contrast, the corresponding BLoCs are not equal:
The phector concept is very important when thinking about physics.
Many calculations can be done in terms of phectors, without being
wedded to one particular basis. Sometimes you don't need any basis,
and sometimes it helps to be able to re-express a given phector in
several different bases. A nifty pedagogical example of this is
the calculation of how much energy it takes to make antimatter, as
we discussed not too long ago: http://www.av8n.com/physics/bevatron.htm
Phectors can be added graphically, by putting them tip-to-tail.
BLoCs can be added numerically, by adding corresponding components ...
provided both summands are based on the same basis; otherwise
they shouldn't be added at all.
Of course BLoCs are very useful, too. In a typical computer program,
the usual strategy is to represent all phectors in terms of BLoCs
using some chosen basis, and then to do arithmetic on the BLoCs.
So, you need to be able to think clearly about both phectors and
BLoCs. Someone who has mastered the subject will be able to go
back and forth from one representation to the other.
I know some people on this list will dismiss phectors as being "too
sophisticated" ... but I don't think that's a valid criticism. In many
ways, the phector idea demands _less_ in the way of sophistication. You
can develop a "gut feeling" about phectors. You can draw pictures of
phector relationships -- angles, lengths, tip-to-tail summation, etc. --
and communicate them to students who are very young, far too young to
understand components, far to young to understand lengths or angles
in terms of dot products.
My impression is that students' proficiency with phectors is a non-
monotonic function of their overall sophistication.
a) They start out with a gut feeling about phectors.
b) Then (usually) this is eclipsed by their understanding of
BLoCs, linear algebra, and all that.
c) Finally when they become really sophisticated, they start relying
on phectors again.
The fact that people heretofore have used the same term -- "vector" --
to refer both to phectors and to BLoCs makes it hard to think clearly
about either one.
I think a valuable and easily-achievable goal is to avoid the eclipse
mentioned in item (b) above ... that is, to teach kids to use linear
algebra while deepening -- not lessening -- their understanding of
phectors as real, physical objects that have meaning independent of
their components, independent of any observers.
I thank Antti S. for motivating me to think more clearly about this.
This is a spin-off of our recent discussion of special relativity,
and partially explains the long-running confusion. Some people were
talking about phectors, and other people were talking about BLoCs,
and we kept talking past each other.
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