Re: i,j,k things.

[Michael Edmiston <edmiston@BLUFFTON.EDU>]

I'm glad this is making some sense, because it took me a while to
figure out what to say.  But, as Leigh Palmer pointed out, it's not
clear that I said it well.

The problem the way I see it goes something like this.
(1) Yes, a unit vector is a vector, hence it has magnitude and
direction.
(2) But, the unit vector is dimensionless, so what does "magnitude"
mean?  We typically like to associate some sort of units/dimension with
magnitude, and here we're not allowed to do that.

Well... when we draw vectors, magnitude is indicated by length.  So
what I'm trying to get at is: how long should I draw this crazy thing.

If the axes we're using are distance and we're talking about position
vectors, it gets especially confusing because it sounds like we're
saying the length of the unit vector is one meter.  That's not what I'm
saying, but that's what Leigh was cautioning about when he made
reference to my comment about using a ruler.

When I draw my graph, especially on a piece of graph paper, the
distance between tic marks that represents a real-world distance of one
meter is nowhere near one meter.  Perhaps one centimeter on my graph
represents one meter in the real world.  That means I will draw my unit
vectors one centimeter long when I put them on my graph.

Perhaps it would be better to use forces (or something other than
distance) for my axes.  So let's imagine that we have drawn three axes,
labeled them as force, put tic-marks, etc. on them.

When we draw the unit vectors i,j,k on this coordinate system, they are
not one newton long.  That is, they don't have units of newtons.  They
are dimensionless.  Nonetheless, when I draw them, they have to be as
many inches, or millimeters, or feet, etc. as we spaced the tic marks
that represent one newton on our axes.  That's why I was trying to say
that we can't really draw the unit vectors (to scale) until we have
defined the axes.

They have to be drawn a certain length so that when we multiply them by
the scalars that are the components of a force vector, the drawn length
of the force vector will come out right.

Stated another way, when I construct the force vector F = 6 N i, the
"drawn" length of that vector on my graph has to exactly contain the
equivalent of six unit vectors stacked end to end.  On my graph the
vector 6 N i might be 1.5 inches long, which means the unit vectors are
0.25 inches long.

So the vector 6 N i has a magnitude which does have dimensions, and
that magnitude is 6 newtons.  But it also has a drawn length along my
x-axis, and that length is six times as long as my unit vectors.  There
is a graphical length to these, but they do not represent any
real-world dimensions.

Leigh, is this wording better?

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:   Saturday, September 18, 1999 2:32 AM
To:     PHYS-L@lists.nau.edu
Subject:        Re: i,j,k things.

What Michael wrote (see below) makes sense to me. It shows that
unit vectors are not defined before the axes and units. Unit vectors
are place holders, j for force is not the same as j for velocity. And,
by the way, I do not remember i,j,k being called versors in Poland,
more than a decade before Hanna. Thanks those who provided
additional good answers.

The thread "Axes not required", started by John Denker, is a
different worth reading topic. At what stage of teaching should
this aspect be emphasized? Why is it important?
Ludwik Kowalski

Michael Edmiston wrote:

> 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. .....