Nothing plotted in any coordinate system makes any sense until we
define the axes. However, once the axes are defined, i,j,k become
defined, and they are vectors.
Example, we first define that we are going to plot all the forces on a
body, and we are going to do this using units of newtons, and we are
going to use a cartesian coordinate system with the origin on the
center of mass of the body.
Once we have said that, then i,j,k become defined vectors. The "thing"
called i has direction (along the x axis) and it has length (length of
one along a newton scale). I believe this is sufficient to make it a
vector... it has magnitude and direction.
The criticism that: [vectors are free to point any direction, and since
i,j,k are constrained to point along their respective axes they cannot
be vectors], does not make sense to me. All it takes to be a vector is
to be something that requires both magnitude and direction to specify
it.
I think this is just a question of semantics. I view the wording "unit
vector" to imply a vector whose magnitude is one as measured by a ruler
calibrated in whatever unit is being used... hence we could call it
"unit-magnitude vector"... but we shorten that to "unit vector."
Also note it is possible to have unit vectors pointing in other
directions as well. It is common, in laws such as Coulombs law, to
define unit-vector-r as having length one and pointing along the line
connecting the two charges and pointed toward the charge for which the
force is being calculated. F12(vector) = (kq1q2/r^2) times
unit-vector-r12. This unit vector is calculated by dividing vector-r12
by the magnitude of vector-r12.
Perhaps your problem is that the word "unit" has more than one meaning.
The principal definition in the dictionaries I looked in was always
"the first and least natural number: One." That is, unit means one.
Your definition of unit (m, kg, N) is also in the definition list, but
not first in the list.
By the way, I agree that the units (m, N) of a vector specified using
unit (magnitude-one) vectors go with the scalar and not with the unit
vector. Unit vectors themselves are dimensionless.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817
-----Original Message-----
From: Ludwik Kowalski [SMTP:KowalskiL@MAIL.MONTCLAIR.EDU]
Sent: Friday, September 17, 1999 11:37 PM
To: PHYS-L@lists.nau.edu
Subject: i,j,k things.
If we agree that in physics the term "unit" refers to things
such as kg, m/s and N, then the term "unit vector" is
probably not appropriate for i, j and k.
Let me elaborate. First, i, j and k are dimensionless,
otherwise they could not be used for so many different
physical quantities. (In F=3i+4j the unit, N, goes with
the scalars 3 and 4, not with i and j.) Second, the
dimensionless quantities, such as v/c, are scalars;
the operation of division does not apply to vectors
pointing in different directions.
So what does the word "unit vector" mean? One unit
of what? F=1j is a vector whose length is 1 N, and
which is directed along the y axis. Likewise F=1i is
a vector. But i and j alone are not vectors. A vector
quantity may have many different directions; j is
always pointing along the +y axis.
I am not proposing to change anything. Only to
recognize that the word "unit" means a different
thing here. Would you agree? We multiply scalars
by i, j and k in order to create vectors. What would
be a better name for i, j and k?
Ludwik Kowalski