Re: [Phys-L] emended puzzle about vectors (and the structure of the world)

[John Denker <jsd@av8n.com>]

In the context of:
  https://av8n.com/physics/img48/vector-sum-difference.png

On 5/4/22 5:49 AM, Chuck Britton wrote:

> I enjoyed it when Martin Gardner included it in one of his
> Mathematical Games columns - Way Back When -

I was not aware of that. Just now I tried to search for
it without success. If anybody knows where to find it,
please let me know.

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

Here's why I find the dot-product construction interesting.
It tells us:
  *If you can measure lengths, you can measure angles.*

  BTW FWIW the converse is not true. If all you have is
  a protractor, you cannot determine length. For example,
  Tycho Brahe super-accurately measured the stars on the
  celestial sphere without knowing how far away they were.

I was reminded of this because I was flipping through Thorne
and Blandford _Modern Classical Physics_. Their eq. 1.4a
defines the dot product without using components, namely:
	A·B = (1/4)  [ (A+B)² + (A−B)² ]
They make a pretty big deal of it, there and elsewhere in
the book.

I should do a book report on that book.

The affinity between lengths and angles is obvious if
you define length in terms of the dot product. I find it
interesting to go the other way, to define the dot product
in terms of length, as in Thorne & Blandford.

It's also obvious if you rely on the metric tensor g. That
is, the squared length of A is g(A,A), and the dot product
of A and B is g(A,B).

OTOH suppose you never heard the term "metric tensor".
It's not a_priori obvious to me that you couldn't have
some other way of measuring length, something that isn't
a tensor, something that isn't bilinear. In particular,
a plain old ruler or digital caliper isn't bilinear. So
it takes some cleverness to use the caliper to measure
an angle. You have to use it twice (i.e. measure A+B and
A−B) and then do some nontrivial data reduction.

I don't know about you, but when I was in high school,
if they had handed me two vectors and a ruler and asked
me to find the dot product *without using components*
I would have had no clue. I certainly would not have
thought of measuring A+B and A−B. Actually, it's much
worse than that; I would not even have understood the
question, since at that stage I thought of the dot
product as being "defined" in terms of components.

It would have been way over my head to think of vectors
as objects unto themselves, independent of whatever
coordinate system (if any) you choose, and independent
of whatever vector basis (if any) you choose.

This reminds me of my first week in college, namely the
first physics recitation section. The graduate teaching
assistant walked in, told us his name, and asked if we
had any questions. We had none. He said "Zo, let me show
someting innntelesting" and then launched into calculating
the energy of the first-ever machine designed to produce
antimatter, i.e. the Berkeley Bevatron. It took a few
minutes, using a dozen lines of algebra, using spacetime
vectors, not using components. I remember thinking that
Toto and I were not in Kansas anymore. I was sitting in
a room full of kids all of whom had been to much better
high schools, and they were equally blown away.

I wrote up the antimatter calculation here:
  https://www.av8n.com/physics/bevatron.htm