Re: [Phys-l] Motion in 1D, vectors and vector components

[John Denker <jsd@av8n.com>]

On 08/13/2007 11:44 AM, Rauber, Joel wrote:

> .... Actually
> introduce vectors purely as geometric objects at first; i.e. arrows,
> this is done briefly, but show adding and subtracting head-to-tail, etc.

That's smart.  That's important.

> Then define velocity pictorially and acceleration.

Yes.


Of course the introduction should be "done briefly" ... but then 
you can spiral back to the arrows again and again, pounding on 
the point that a vector is a physical, geometrical object, not 
just a list of three numbers.

As it says at:
  http://www.av8n.com/physics/two-vector.pdf

Paraphrasing a bit:

Students’ proficiency with geometrical vectors seems to be a 
regrettably non-monotonic function of their overall sophistication:
  a) They start out with a gut feeling about geometrical vectors.
   In early grades they learn to draw vectors as arrows. They add
   them graphically, tip-to-tail.
  b) Then they learn about lists of components, matrices, and all that. 
   They add components and matrix elements numerically, component by 
   component. (This assumes the the representations are all based on 
   the same basis;  otherwise they shouldn't be added at all.) So far, 
   so good. 
   The problem is that all too commonly, their knowledge of components 
   eclipses their geometrical understanding of vectors.
 c) Finally when they become really sophisticated, they start making 
  use of geometrical vector concepts again.  There is an important
  difference between a tensor and the matrix elements of that tensor.

A valuable and easily-achievable goal is to avoid the eclipse mentioned 
in item (b).  That is, we should teach people to use components while 
deepening – not lessening – their understanding of vectors as real, 
physical objects that have meaning independent of their components, 
independent of any basis, and independent of any observers.


================

Possibly constructive suggestion:  I find it useful to distinguish:
 -- the x-component of a vector (which is a scalar) versus
 -- the x-projection of a vector (which is a number).

In particular, 
 -- the x-component of V is <x|V>            (a scalar)
 -- the x-projection of V is |x><x|V>        (a vector in the x-direction)

where |x> is a unit vector in the x-direction.

  I realize that some people promiscuously use "component" to mean
  either the scalar /or/ the vector (i.e. the thing I am calling the
  projection).  However, I think it is worth making the distinction.
  If somebody has better terminology, i.e. a better way of making
  the distinction, I'd love to hear about it.

Projections have the nice property that we can take projections 
in any direction, not just along some pre-ordained set of basis 
directions.  In general, the projection operator in the q-direction 
is

             |q><q|
            --------
             <q|q>

for any nonzero vector |q>.  This allows you to choose a |q>
that is _physically relevant_ to the problem, as opposed to
a pre-ordained basis that typically has little or no physical
significance.

As a rule, anytime you can formulate the problem geometrically,
using projections, you're better off doing it that way, rather
than introducing a basis and grinding out the components.

The physics is in the arrows, not in the components.

There are eleventeen reasons for emphasizing the geometrical
approach, not least of which is that it lays the foundation for 
understanding relativity in terms of spacetime and 4-vectors.
This approach is simultaneously simpler /and/ more sophisticated.



    Tangential remark:  It is important to keep computers in
    their place.  When computers work with vectors, they use 
    components internally ... but so what?  The physics is 
    still in the arrows, not in the components.  By way of 
    analogy, computers do arithmetic using binary internally, 
    but that doesn't mean we humans should switch to using 
    binary for everyday purposes.  I've never seen a 
    110111 MPH speed-limit sign.  (I keep expecting some
    smart-aleck student to make one, but it hasn't happened
    yet AFAIK.)


When it comes to geometrical vectors, some textbooks are better 
than others.  Some of them start out OK, defining vectors as 
arrows ... but then lapse into component language as soon as 
they start doing actual problems.  Fooey.

If you are in a hurry, you can judge a textbook according to 
how it defines the dot product. 
 -- |A| |B| cos(theta),  or
 -- Ax Bx + Ay By + Az Bz

You can guess which version I prefer.  

A second quick check involves looking to see if the term
"projection" is in the index.  Alas I don't offhand know of
any general-physics texts that pass this test.  (If anybody 
knows of one, please tell us about it.)

A more thorough check of the text involves looking at the
end-of-chapter problems to see if they involve geometric
relationships between vectors, as opposed to grinding out
components.