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



On 08/16/2007 04:55 PM, Rauber, Joel wrote:

How about:

Component vector => again I like this since it is self-describing

I have no objection to that, although IMHO "projection" is
more concise, is equally self-describing, and is already
well-established in the math and physics literature.

Projection is an idea (and a term) worth knowing anyway,
so it seems like a win/win to use it in this context.

Component value => probably removes some ambiguity and is one word less
than vector component value

How about "element"? The 100% conventional name is "matrix
element" but for an introductory physics course it might
be advantageous to shorten it to "element", just because
the m-word scares people.

I'm still looking for the holy grail of terminology here . . .

Me too. I think we are now closer to finding it than we
were a week ago.

My current best shot at all this is at
http://www.av8n.com/physics/intro-vector.htm
especially
http://www.av8n.com/physics/intro-vector.htm#sec-projections-elements
which has changed quite a bit since a week ago.


It says in part:

Possibly constructive suggestion: It is useful to distinguish:

* a matrix element, versus
* a projection.

Depending on context, the word “component” can refer to either
a matrix element or to a projection, as follows:

* A matrix element can be considered a component of an array.
Each matrix element is just a number. If you know all the
matrix elements of a given array, you can reconstruct the
array by putting all the matrix elements together in a list,
in order.
* A projection can be considered a component of a geometrical
vector. Each projection is a vector unto itself. If you
know all the projections of a given vector onto a complete
set of orthogonal directions, you can reconstruct the vector
by adding the projections together, using the tip-to-tail
rule.

For any geometrical vector V, this means:

* the x-matrix-element of V is the scalar ⟨x|V⟩, while
* the x-projection of V is the vector |x⟩⟨x|V⟩.

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

Remember that the term “component” could mean either the scalar
⟨x|V⟩ or the vector |x⟩⟨x|V⟩. If you need to avoid ambiguity,
you can avoid the term “component” entirely; if you mean “matrix
element” say “matrix element”, and if you mean “projection” say
“projection”.


The physics is in the arrows,
not in the matrix elements.