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



-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Monday, August 13, 2007 12:54 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] Motion in 1D, vectors and vector components

/Snip/

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 vector).
***Minor edit to last line above: last word changed from "number" to
"vector" to reflect correction made by author.

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.


Thomas Moore in his book /Six Ideas That Shaped Physics/ uses the
terminology "vector component" for the value-with-units and "component
vector" for what you call a projection.

I think the word "projection" is, in general, used for both what you
call a component and what you call a projection. For a couple of
examples where it is used to mean the number-with-units see:
<https://carnot.physics.buffalo.edu/archives/2002/09_2002/msg00392.html>

where it says:
" 'component' = as Bob says below, the projection of the vector of
interest onto the axes, what is traditionally labeled A_x and A_y
(signed scalar quantities)."
or see:
<http://www.av8n.com/physics/acceleration.htm>
where it says:
"The scalar acceleration can be considered one component of the vector
acceleration, namely the projection in the 'forward' direction (although
this is undefined if the object is at rest)."

Perhaps the use of the following terms would help avoid confusion:
vector component value
component vector

Note: I hesitate to use the word "scalar" for the vector component
value. I don't think of a vector component value as something that
transforms as a scalar (remains the same), for instance under rotation.
The new x component of a vector after the transformation is not, in
general, the same as the old x component. One could argue that when you
transform from an unprimed frame to a primed frame what you mean is that
the x' component is not equal to the x component but that the scalar
product of the vector with the original x-axis basis vector still has
the same value so the x component value is indeed invariant under
rotation. But if one is going to use that argument then it would seem
that one could argue that the time-like component of the four momentum
is invariant under boosts, as long as we mean the component along the
original time-like axis. At any rate, I blame Feynman for my
hesitation. Up until a couple of weeks ago, I would have been perfectly
happy calling the x component value of a vector, a scalar. Then I read,
on page 2-4 of Volume II of the Feynman Lectures on Physics (where the
derivative in the quote is actually a partial derivative):

"What is the derivative of a scalar field, say dT/dx? Is it a scalar,
or a vector, or what? It is neither a scalar nor a vector, as you can
easily appreciate, because if we took a different x-axis, dT/dx would
certainly be different."


/snip/

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.)

/snip/