It appears to me there is some confusion between (1) the convention of
how axes are labeled and angles are measured in a standard coordinate
system, and (2) how the coordinate system is oriented.
The convention that an xyz coordinate system has mutually perpendicular
axes, that we choose to use the right-handed version (such that i cross
j = k), that angles in the x-y plane are measured counterclockwise from
the x axis when the origin is viewed from a position on the positive z
axis...etc... has nothing to do with how we choose to orient that
system.
* * * off on a tangent * * *
If there is no good reason to choose otherwise, I prefer to orient the
coordinate system so positive-x is to my right, positive-y is in front
of me, and positive-z is up.
Many textbook authors seem to have an aversion to drawing the positive-y
axis into the page. Thus, to maintain a right-handed system they draw
positive-y to the right, and positive-x coming out of the page. This
means motion forward is in the negative-x direction. I personally
don't like forward motion expressed as a negative number.
The way I orient the coordinate system, motion to my right is positive,
forward motion is positive, and upward motion is positive. If this is
what I want, and I also want the positive z-axis to be up, then there is
only one way to draw a right-handed coordinate system to acheive these
goals.
However... remember that I began this "off on a tangent" with the words
"if there is no good reason to choose otherwise."
* * * back on track* * *
Each of us can choose what coordinate system to use, and how to orient
the chosen coordinate system. But once we choose a coordinate system
and its orientation, we ought to follow conventions for using that
coordinate system.
Let's take an example. Let's not choose the standard right-hand xyz
coordinate system. Instead, let's choose the NSEW coordinate system
used by cartographers. Let's imagine we will draw a map and plot a
course for someone to take. Let's not worry about magnetic corrections.
Do we have to put N (north) at the top of our page? Certainly not. We
can put N anyplace we want; architects, surveyors, etc. do this all the
time. An option on my Garmin GPS is to have the computer orient the map
so the direction of travel is at the top of the display. This is called
"track up" as opposed to "north up.". Some people prefer track up
because it helps them visualize better whether something is to their
right or left or in front of them or behind them. Therefore, if "track
up" mode is selected, N could be anywhere on the display depending upon
which direction you are currently travelling.
Let's put N on the right side of the page. Okay, having put N on the
right side of the page, am I free to put E at the top of the page?
Well... I can do it, but it would be pretty dumb because it is not
convention and it is bound to lead to confusion.
* * * tangent * * *
Likewise, I ought not draw an xyz coordinate system with x to the right,
z up, and y coming out of the paper. Why... because it is not a
right-handed system, and unless there is good reason to choose a
left-handed system, people have agreed to use right-handed systems.
* * * back on track * * *
Okay, so we have oriented our map with N to the right and E at the
bottom, etc. We want to direct the travellers to a location 32 miles
away, and the location appears on our paper map near the top of the
page, but a little to the right of straight toward the top.
Should we tell people to travel 32 miles at bearing 75-degrees? I hope
not. They won't end up at the destination. They'll end up near the
bottom of our map page because they are assuming 75-degrees is clockwise
from N. Therefore we should direct our travellers to go 32 miles at
bearing 285-degrees.
Would it be okay to tell our travellers to go 32 miles with a bearing
that is 15 degrees N of W? It would be okay, but our travellers won't
be able to read that number directly from their compass or GPS, because
when they are heading 15 degrees N of W their compass/GPS will say 285.
* * *Major Point* * *
I hope my map/navigation example demonstrates that it is pretty
important to follow conventions when you are drawing maps and giving
directions. If we want to be understood, we don't really have the
freedom to redefine NSEW nor to redefine what bearing means. We do,
however, have the freedom to draw a map with any orientation we want, as
long as we mark which way is north.
Likewise, we can orient a standard right-handed xyz coordinate system
any way we want, but once we decide the position of the origin and the
orientation of one of the axes, the locations of the other axes become
fixed, and the angles and distance to a particular spot become fixed.
If we don't follow that, then we risk miscommunication. Additionally, I
maintain that following the convention also makes it easier for
students.
Some have been writing that they want their students to realize that
angles are relative to our choice of zero. I say, "exactly right." But
then these writers go on to say such things as it isn't necessary to
measure angles counterclockwise from the x-axis. To this I say, "Did
you draw an x-y coordinate system on your picture? If so, did you
orient the x-axis along your choice of zero angle? If not, why not? In
fact, if you chose your zero-angle reference line and then didn't draw
the x-axis in that direction, then why did you bother to draw an x-y
coordinate system at all? What was your point? If you're not going to
use the coordinate system, why did you draw it?"
I give a fairly long list of suggestions to students for analyzing net
force and Newton's second law. The initial items on my list of
suggestions goes like this...
(point 2 in my list)Draw a free-body diagram.
(point 5 in my list)Choose a coordinate system. Generally, choose one
axis to line up with the motion or the intended motion or the suspected
motion. Although this is not necessary, it usually makes the vector
algebra easier. For example, this usually means the coordinate system
should have either the x-axis or y-axis aligned with the surface of the
incline if the object is sitting or sliding on an incline.
Once students have done this, they have chosen the zero-angle direction
by their orientation of the x-axis. From that point on, I expect them
to measure angles from that reference toward the y-axis.
Note, the reason I said "toward the y axis" rather than counterclockwise
is because counterclockwise is ambiguous in a 2D coordinate system. A
student could aim the x-axis to the right and the y-axis down. In that
case, from our viewpoint (which would be a viewpoint from the negative
z-axis) the positive angle direction would appear clockwise.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu