On 10/03/2003 07:53 AM, Justin Parke wrote:
> Specifically, they have trouble when one or both (we are working in
> 2D) of the components of the vector are negative in which case simply
> taking arctan (y/x) does not yield the proper angle.
As a practical matter, note that most computer languages
have a function atan2(x,y) which prevents all sorts of
headaches. It does the right thing in all four quadrants,
returning values in the half-open interval (-pi .. pi].
It agrees with atan(y/x) in quadrants I and IV. It fails
if x and y are *both* zero, but works fine otherwise.
Even excel (not noted for its mathematical sophistication)
implements atan2(,). Some hand calculators implement it;
others do not.
====================
On the philosophical/pedagogical level: How about this:
Show of hands:
-- How many of you use a spoon sometimes, for eating
soup and such?
-- How many of you are using a spoon right now?
Good, that means you realize that something that
is appropriate some of the time may not be
appropriate all of the time. Part of the story
is not necessarily all of the story.
-- How many of you think that
direction(V) equals atan(Vy/Vx) SOMETIMES?
-- How many of you think that
direction(V) equals atan(Vy/Vx) ALWAYS?
Using atan(Vy/Vx) is like a spoon. Sometimes it's
the whole solution, sometimes it's only part of the
solution, and sometimes it's not appropriate at all.
For instance, if V=[0,1] then the direction of V is 90
degrees, and the atan() function plays absolutely no
role in the calculation.
Get used to it: In this class you will learn lots of
formulas that are like spoons. You have to learn the
formula *and* learn the limits of its validity.
================
The five-step approach presented by Michael E. makes
a lot of sense.
I would only add that the students should know by sight
the eight cardinal directions: zero, 45 degrees, 180,
135, et cetera. They should be able to give the number
given a sketch (e.g. flashcards) without a calculator.
Conversely they should be able to instantly sketch any
of the eight cardinal vectors given the number.
=========================================
An additional source of confusion: If you get students
who have experience in navigation, they may be accustomed
to measuring angles clockwise from north. Physics and
math books generally measure angles CCW from east.
On 10/03/2003 11:39 AM, RAUBER, JOEL wrote:
> However, I tell my students I'll accept [anything]
> 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.
>
> If they don't tell me I tell them I'll grade according positive
> angles being measured CCW from the positive x-axis.
Right on!
============
There's also issues about whether V=[1,-1] should be
represented by direction 315 degrees or -45 degrees.
In some contexts one or the other is strongly preferred.
-- Compass headings are always expressed in the range
(0 .. 360]
-- Calculators generally spit out something in the
range (-pi .. pi] whether you like it or not.
I would hope students would recognize 315 and -45 as
being synonymous.