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

[John Denker <jsd@av8n.com>]

On 08/17/2007 10:56 AM, Jeffrey Schnick wrote:
> ... I think I would rate "array element" higher than
> "matrix element".  

I can live with "array element".

I suspect for the purposes of an introductory discussion
of vectors, shortening it to "element" would be just fine
... in which case we don't need to ask whether it is short
for "matrix element" or "array element".

> I believe I have this prejudice because my first
> encounter with the word "array" was (perhaps) in a programming
> context where the first kind of array I encountered was a
> one-dimensional array whereas my first encounter with the word
> "matrix" was (perhaps) in a mathematics course where the first kind
> of matrix that was referred to as a matrix was a two-dimensional
> matrix.

Yeah, but there's a lot of good things you can do with non-square
matrices.  For starters, it's good to be able to distinguish row
vectors from column vectors, i.e. to distinguish one-forms from
pointy vectors.  That shows up in QM (bras and kets) and in thermo
and in general relativity.

> A Google search on "scalar projection" indicates that some people
> think that there are at least two kinds of projections of a vector
> onto another vector.  See, for instance, 
> <http://en.wikipedia.org/wiki/Dot_product> where a diagram caption
> reads "|a|*cos(θ) is the scalar projection of a onto b". 

Ouch, that expression hurts my ears.  It's going to take more
than some unsourced wikipedia article to persuade me to adopt
that terminology.  And there's no need for it;  it would be
just as easy to say that |A|cos(θ) is the /length/ of the 
projection of A onto the direction of B.


> This might
> actually argue in favor of using the word "projection" for what some
> folks call a "component vector" in that it would help folks learn
> what a projection really is but it indicates that it is not
> self-describing to everybody. 

None of these terms are self-describing to naive students.
You can't talk about "orthogonal" projections in front of
students who don't know what "perpendicular" means, let 
alone what "orthogonal" means.

I think we need to base most such decisions on a certain
amount of lookahead:  we should decide what ideas we want 
to /become/ established in the students' minds.  Projection 
operators are used all over the place in physics, and 
projective geometry is a respectable subfield of mathematics.

In the circles I move in, the idea of projection gets used
many times per day.  Consider the pin-connected strut:  it
is the most natural thing in the world to ask what is the
projection of the strut-force Fs onto the direction of
motion of the anvil.

For me, it's a cost/benefit analysis.  The cost of introducing
the idea of "projection" is no greater than the cost of any 
other way I've seen of solving this problem ... and the payoff 
is much higher.

> I like component vector because it is
> self-describing; its name tells everyone that it is a vector.

Yeah, but it's not as useful as the "projection" idea.

Projection is a physics /idea/ ... not mere terminology.

Ideas are primary.  Terminology is secondary.