John Mallinckrodt says he disagrees with my suggestions about drawing
vectors. But I believe he actually agrees with me for the most part.
He says, "I would say draw them any length that is convenient; your
only obligation should be to draw them the *same* length." And in
other parts he talks about scaling vectors with respect to each other
if they are measuring the same property (force, for example).
I say, yes, that's my point exactly. On the same drawing (let's limit
it to two dimensions), if we want to draw one velocity vector that is 5
m/s at 30-degrees, and also another velocity vector that is 8 m/s at
70-degrees, then we can draw those two vectors any way we want as long
as (1) their drawn lengths are in the ratio of 5:8, and (2) the angle
between them is 40-degrees.
Of course you don't even have to do that if you don't want to, but in
that case you ought to warn your audience that you are not drawing your
vectors using the same scale.
If you HAVE drawn them using the same scale, then if you also decide to
draw a unit vector on the same graph, and if you also want it to be
drawn to scale, it has to be one-fifth the length of the 5 m/s vector.
But be sure to realize that I only apply this statement to a plot in
which the axes are calibrated in velocity units. This does not apply
if the axes are distance, or force, or something else.
If you reread my statements about drawing vectors you ought to notice
two things. (1) I was always assuming we are interested in drawing
vectors to the same relative scale. (2) I was giving examples of
vectors drawn on a coordinate system in which the axes indicated the
same unit as the property being graphed.
It is certainly possible, and we do it all the time, to draw vectors on
a set of axes for which the axes represent something different than the
vectors being drawn. For example, we often draw a circular path on an
x-y graph to represent circular motion. Then we draw velocity vectors
tangent to that circle and we draw acceleration vectors exactly
centripetal (if uniform motion) or skewed from centripetal (if we have
non-uniform circular motion).
In this case the velocity vectors can be drawn any length as long as
they have the same relative scale to each other. A 5 m/s velocity
vector ought to be half the length of a 10 m/s velocity vectors.
Likewise, the acceleration vectors can be drawn any length they want
as long as they are in the same relative scale (with each other...
which does not have any bearing on the relative scale used for the
velocity vectors).
If anything I said in an earlier posting would be construed to mean
something different than this, it was surely not intended. My whole
goal was to try to bring home the point that the drawn vector length is
not TOTALLY arbitrary unless we do not intend to draw them to scale.
And I totally realize that "drawing vectors to scale" is something
that is done within a set of vectors that signify the same property.
Velocity vector lengths are drawn to scale with each other, they have
no explicit or implied scaling with respect to force vectors.
BTW, I do not understand Johns examples of 6 kg, 2 kg, 2 hp, and 3000 W
vectors. I always thought mass and power both fall in the category of
scalars and therefore would never be drawn as vectors.
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