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[Phys-l] how to understand vectors



On 11/09/2010 12:50 PM, treborsci@verizon.net made several
excellent points, each of which would be worthy of a thread
of its own.

I have always handled the plus or minus sign problem in one dimensional
situations as follows.

1) Begin with an introductory (just addition and subtraction) exposition of
3 dimensional (Cartesian) vectors. Introduce and freely use the unit vectors
I, j and k.

2) Impart some facility by working examples in two dimensions,
geometrically and algebraically. Explicitly use unit vectors.

Right on, brother.

I have not much to add, except to underline the the importance of
doing things graphically and geometrically. I see it as a three-
stage developmental process:
a) Early on, students are given a definition of vector as having
a magnitude and direction.
b) Later they learn how to represent vectors as a list of numbers.
c) Eventually (maybe in graduate school) they learn that the truly
sophisticated approach is to treat vectors as geometric objects,
independent of any basis and therefore *not* fundamentally lists
of numbers. So they need to go back and re-learn the geometrical
and graphical ideas they used in step (a).

I mention this because life would be better if in step (b) students
could learn about components while continuing to see things in terms
of magnitude and direction, continuing to add vectors graphically
and geometrically (tip-to-tail), et cetera.

I like to emphasize that the list of numbers means nothing without
the basis vectors. I like to be specific about what basis is being
used, using an "@" sign suffix. For example we might find that

[ 1 ]
v = [ 0 ]
[ 0 ] @ Moe

is the same vector as

[ 0 ]
v = [ 1 ]
[ 0 ] @ Joe

given that Joe's basis is rotated relative to Moe's. Different
components, same physical vector.

I would also remark that some computer languages use the word "vector"
to apply to any list of numbers, even if the "components" of the
"vector" don't even have compatible units or dimensions ... even if
they are not even numbers. I'm too open-minded to say one usage is
right and the other is wrong, but I will say they are inconsistent,
and this creates a big risk of negative transference.

3) Go to one dimension, working examples always using the unit vector i.
Use vector addition and subtraction, geometrically and algebraically.

Yes indeed.

4) Carefully notice that in one dimensional algebraic expressions the unit
vector is superfluous (it can be cancelled out of any equation, or taken out
of any expression as a common factor). The only directionality to be
specified is forward or backward, and that is just as unambiguously handled
by signed numbers ( + vs - ) as by signed unit vectors( +I vs -i). So in
the interest of conservation of ink/chalk, we can drop the unit vectors from
one dimensional vector situations and simply use the convenient notation of
signed numbers. Emphasize that the unit vector (i) is still implicit (as an
overall common factor) in these one dimensional expressions. Beginners
might be encouraged to still explicitly write each one dimensional vector as
a magnitude with a signed unit vector - until it becomes "natural" to infer
the absent unit vector.

Yes indeed.

I would add that writing the unit vector explicitly is a good idea
in general, not just for beginners. There are many situations where
writing the unit vectors is a very good idea, and the experts know
it is a good idea, even if the beginners don't.

A classic example concerns one-dimensional motion around a closed
path. This could be a train on a track, or current in a wire.

The following two circuit diagrams are not equivalent:

|------------>-----------|
| I |
| |
|------------------------|


|------------------------|
| |
| I |
|------------>-----------|

In one diagram, positive current I flows clockwise, while in the
other it flows counterclockwise. The ">" symbol is in effect a
unit vector specifying the direction of positive current. The
actual physical current is a vector, namely I times this unit
vector.

Similarly for a train on a track, the motion is one dimensional
but there is no good way to specify whether positive motion is
CW or CCW ... except by reference to a unit vector of some kind.

Bottom line motto: Express the unit vectors explicitly, unless
you are really really sure you don't need to. Even if you "could"
conceal them, that doesn't mean you need to or want to.