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Re: vector directions



Mike wrote in part (interspersed with my comments):

Well... what is it that we are trying to teach? Are we teaching that
"anything goes" or are we teaching that one of the goals of learning
science is learning common practice so as to make it more
likely that we
can communicate clearly.

Both! If I interpret "anything goes" as physics is independent of coordinate
system. I try to make my students aware of common practice, hence the
statement about how I'll grade their answers if they don't tell me they are
deviating from common practice. I.E. if you deviate from common practice,
make sure your reader is aware of it! (IMO, that is an important lesson.)


If we are converting from i-j notation to r,theta notation, or
converting from i-j-k notation to r, theta,phi notation (or
vice-versa)
there are some conventions we assume we are using. For example we
assume orthogonal axes and we even assume a right-handed coordinate
system.


(though math books often use the reverse convention for theta and phi) &
electrical engineers use j instead of i for square root of minus one . . .
ad infinitum

. . .


This is the convention in math books, physics books, high-school or
college, and also what practicing physicists use in publications, etc.
Why wouldn't we want students to understand this and use this
convention?

Of course, I want my students understand prevalent conventions and what to
do if they desire to deviate from them. And that if they find it more
convenient to work with a different convention there is certainly nothing
wrong with doing so.

For Vector Components: IMO, slavish use of the measure angles CCW from the
+x axis (or in 3D, use the standard physics convention for spherical polar
angles) often isn't the easiest route for working out the geometry of a
problem. And I want my students to understand that as well.

Demanding use of such a convention, sometimes has the unfortunate affect of
causing students to memorize x-component equals hypotenuese times Cos
(theta); which they blindly apply to problems, even if the given angle theta
is such that it would be more appropriate to find the component as
-Sin(theta). Not to mention the confusion in some students minds when you
choose your coordinate system axes to not correspond with horizontal and
vertical or if you wish to find a component of a vector along an odd
direction.

Nothing beats a diagram to keep things straight. I try (but often fail) to
get students to draw diagrams and use whatever trigonometry is appropriate
for the given information to find the unknown information. One can always
use the diagram to convert your answers at the end to be in terms of common
convention rather easily. This emphasis gets at what is important about
vectors (the vector nature of certain quantities) in a better fashion than
memorization of conventions for calculating components. IMO.


Yes... we also want to teach that physics is independent of the
coordinate system used. Yes, we don't really care which way
the student
orients the axes as long as they maintain a right-handed system. But
once a standard coordinate system has been chosen and the orientation
has been chosen, we ought to follow conventions.

I do not totally disagree with Joel's assertion that we can accept any
convention as long as the student makes it clear what convention is
being used. Clarity is a very significant goal in our
teaching. But if
clarity is what we are after, and assuming this implies accuracy and
ease of communication, then we also ought to teach (and
expect students
to be able to use) the conventions that are fairly well established.

I think we basically agree in the main, but quibble on the importance of
"convention" vs "independence of coordinate system" and possibly disagree as
to how well established some specific conventions may be.

Incidently, I wouldn't mind if a student used a left-handed coordinate
system, if they did so correctly and consistantly and informed me the reader
that they are doing such. When I introduce cross products I take advantage
of the topic to distinguish between left-handed and right-handed coordinate
systems. And usually give a warning that sometimes you will see British
authors use left-handed coordinate systems. I've been bitten by that one in
using some British reference. I suspect that the left-handed practice in
Britain is significantly less than 50 years ago. Can anyone on the list
shed light on that comment?

Joel R.

PS, my comments are based on what I do for calculus level university
physics and more advanced courses. For a high school course, I would likely
adhere more closely to Mikes comments about standard practice, both for the
reasons that Mike mentions as well as for reasons of providing a very strict
algorithmic approach for such students. (this naturally depends on the
abilities of your high school students)

PPS, nothing beats a decent drawing!