Re: i,j,k things.

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Thu, 16 Sep 1999, Michael Edmiston wrote:

> ...
> 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.

I would say draw them any length that is convenient; your only obligation
should be to draw them the *same* length.

> ...
> 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.

Frankly, I'm surprised to hear this suggestion.  Imagine how inconvenient
life would be if we were drawing a representation of the solar system and
felt obliged to draw unit vectors with a length equal to that which
represents one meter. In my opinion, it is grossly misleading to imply any
connection whatsoever between the length of a unit vector and the distance
that you are using to represent one meter graphically.  The length of a
unit vector is "one", NOT "one meter", not one astronomical unit", not
"one Newton", not "one second", not "one kilogram", etc.

If you have drawn and scaled axes that graphically represent space, then
you are obliged only to represent vectors with dimensions of length
according to your chosen scale, but vectors of dimensions *other* than
length (like unit vectors, velocities, electric fields, etc.) can each be
scaled arbitrarily and independently.  For instance, a vector with a
magnitude of 6 kg can be represented as having any convenient length, but
should be three times as long as one with a magnitude of 2 kg and,
regardless of the choice made for vectors with dimensions of mass, a
vector with a magnitude of 2 horsepower can again be represented as having
any convenient length but should be approximately half as long as one with
a magnitude of 3000 watts.

> 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.

Right.

> 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. ...

No, no, no!  (No?)  Again I am genuinely surprised to hear this
suggestion.

> 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.

When I multiply a unit vector by a scalar with the dimensions of force,
then I have a force vector and it need only be scaled in proportion to
other force vectors.  How would you propose to draw in a single graphical
representation the forces, displacements, velocities, and accelerations
involved in a scenario like the following: a set of forces with magnitudes
on the order of 100's of kilonewtons are applied to a large object with
dimensions on the order of 10's of meters producing an acceleration on the
order of (millimeters/second)/second until its speed is on the order of
kilometers/second.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm