To paraphrase what Timothy Folkerts wrote at 04:39 PM 9/21/99 -0500:
When working with vectors in, say, velocity space,
-- a unit vector has magnitude 1.0 (which is the multiplicative identity).
-- a unit vector does *not* have magnitude 1.0 meter/second (which is the
unit of measurement).
He asks:
Am I in the minority???
To which the answer is:
"no, but it wouldn't matter even if you were, because you are right".
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This gives me a new way of looking at what Ludwik Kowalski wrote on several
occasions suggesting that unit vectors are somehow different from other
vectors. They are undoubtedly vectors --- but they are not vectors with
dimensions of velocity. So if all the other vectors in the problem are
dimensionful velocity vectors, we have to be careful.
In particular, on a graph where we have drawn dimensionful velocity vectors
(or any other dimensionful vectors), we can also draw a unit vector -- but
we must realize that it lives in a "parallel universe" and any comparison
between the as-graphed length of a dimensionful vector and the as-graphed
length of a dimensionless vector is meaningless.
Obviously, graphing two parallel universes on the same piece of paper can
lead to confusion. If you don't warn them, some people will try to make
the forbidden comparison of the as-graphed lengths. So warn them already!
And remember the very old saying:
abusus non tollit usum
i.e.
misuse does not nullify use
or in more colloquial terms,
no matter what you're doing, you can always do it wrong.
That doesn't prevent other folks from doing it right.
In proper hands, drawing parallel universes on the same piece of paper is a
fine thing to do. In particular, you *must* do this if you want to
represent a velocity field: At point X you draw a velocity V. The
position vector (X) lives in different universe from the velocity vector
(V), and any comparison between their as-graphed lengths is invalid. You
can *independently* choose the units of measurement for each universe.
---------------------
There is another way of approaching the issue which *does* allow us to
speak of unit vectors in velocity space: we just use dimensionless velocities.
In relativity problems, we often choose to measure all velocities as
multiples of the speed of light. In acoustics problems, we often choose to
measure velocities as multiples of the speed of sound.
The main downside to such a choice is that you partially lose the chance to
cross-check your calculation using dimensional analysis. But nothing
requires such a cross-check; if you calculate carefully you don't need it.
Specifically, I think that John Mallinckrodt got it exactly right when
at 05:57 PM 9/21/99 -0800 he wrote:
0>
1> I think texts set a HORRENDOUSLY bad example when they write things
2>like, "The position of a ham sandwich is given by x = 3 - 5t + 2t^2,
3>find its speed at t = 5 s."
because line 3 tries to stick units onto something that was dimensionless
on line 2.
But once again
abusus non tollit usum
and the repaired statement
% "The position of a ham sandwich is given by x = 3 - 5t + 2t^2,
% find its speed at t = 5."
is a perfectly reasonable example of using dimensionless time and distance
variables.
______________________________________________________________
John S. Denker jsd@monmouth.com