Re: vector directions

[Robert Cohen <Robert.Cohen@PO-BOX.ESU.EDU>]

This is an interesting discussion.

JOEL_RAUBER@SDSTATE.EDU writes:
> However, I tell my students I'll accept any of the angles mentioned 
> above if they tell me unambigously how they are measuring 
> the angles 
> (relative to what and what they are calling positive vs. negative 
> angles) and if their answer matches their fiducial statements.

to which Justin Parke responded:

> I tell my students the same thing but many of them need an 
> absolute recipe as they are not yet comfortable with the 
> mathematics involved.

I don't want my algebra-based college-level students to do
vector analysis (or anything) according to an absolute recipe,
even if it means they get the right answer.  Rather, I want
them to see that angles are relative and that our choice of 0
(and positive) is arbitrary except for the local conventions.
Indeed, some problems are easier if 0 is off at an angle
(e.g., if we already know the direction of the acceleration
and we want to find the net force, we might as well choose
0 as the direction of the acceleration).

Consequently, this semester I tried something new.  I first
explored the relative nature of angles.  For every problem, I
have them choose their own 0 and 90 degree directions.  Once
they have chosen their 0 and 90 degree directions, they use
sines and cosines to get the 0 and 90 degree components (which
we call x and y by convention) and to get the direction from the
components, they use inverse cosine.

It all seemed to work real well except for one little problem:
many of my students couldn't calculate the angles from any
arbitrary direction.  For example, I gave them the following
problem in class the other day:

Suppose we are given a direction of 40 degrees counter-clockwise
from East (where North would be 90 degrees counter-clockwise from
East).  Which of the following are equivalent to this direction?

A. 140 degrees clockwise from West
B. 50 degrees clockwise from North
C. 130 degrees counter-clockwise from South
D. All of the above
E. None of the above

Less than half chose D (I know the numbers because we use CPS
remotes in class).  This is after using vectors for at least a
week.

It took some peer teaching before most of the students understood
how to measure angles from various reference directions.

The problem was that I assumed students could do this.
I was wrong.  No wonder they do better with an absolute
recipe.

Next time I'll know to explore this with the
students before talking about vectors and components.

____________________________________________________
Robert Cohen; 570-422-3428; www.esu.edu/~bbq
East Stroudsburg University; E. Stroudsburg, PA 18301