Re: vector directions

[Aaron Titus <titus@MAILAPS.ORG>]

I'm catching up on email, so if this has already been answered in a
similar way, then I apologize. (I haven't read the rest of my 70
messages yet)

I had always been frustrated with my teaching of vectors and student's
learning of vectors. I think that the frustration is due to various
things, one of which is the way I presented it. However, I made some
changes to how I present and use vectors throughout the course. Some of
these changes were made because of my frustration and others were made
because of the textbook I now use, Matter & Interactions. Here are some
general recommendations with some commentary:

(1) Vectors should be taught in context, not as general mathematical
constructs.

I begin with position vectors. How do we specify the location of
something? (using the Cartesian coordinate system of course). Then I
discuss how we specify the "movement" of something from one location to
a second location. This brings up the idea of displacement (r2-r1) and
how to subtract vectors, both algebraically and pictorially.

This leads easily into average velocity.

We also discuss relative position, the position of object 1 with
respect to object 2.

(2) Vectors should be taught in 3-D with examples and exercises that
include 2-D and 1-D.

Only doing 1 or 2-D examples hurts students, I believe. I especially
resist the traditional method of teaching 1-D motion before 2-D. If you
start with 3-D, then many difficulties students have with transitioning
to 2 or 3-D from 1-D are avoided.

I'll never forget a student looking at a 2-D velocity vector and asking
if it was positive or negative. This was after two weeks of
instruction, 1 week on 1-D and 1 week on 2-D. That student's question
completely changed how I teach. I saw clearly the difficulty students
have when studying 1-D motion and then being thrown into 2-D. It's not
an easy transition and it's unnecessary I think.

(3) Vectors should be specified in a way that is consistent with how
vectors are specified in calculus courses and in the TI calculator.

Writing a vector in component form, such as A=<Ax, Ay, Az>, using the
notation of < > or [ ] is preferable to unit vector notation. I've
found that using this notation, my students RARELY make the mistake of
adding vector components as scalars and specifying a vector as a
number.  This mistake was COMMON when I used unit vector notation
however.


(4) When calculating angles, find the angle with respect to the x-axis,
y-axis, and z-axis as necessary. These angles (alpha, beta, gamma for
angle with respect to +x, +y, +z respectively) are sometimes called the
direction cosines:

Ax=Acos(alpha)
Ay=Acos(beta)
Az=Acos(gamma)

These will always give the correct signs (or correct angles depending
on which one you're solving for). Another advantage of teaching this is
that students will see it in future math and science courses.

I highly encourage you to look at the Matter & Interactions treatment
of vectors at

http://courses.ncsu.edu:8020/py205/lec/004/Docs/Handouts.htm

Also see the Velocity Notes handout. The only thing I do significantly
different is that I teach the "direction cosines" approach to angles.


Aaron


On Friday, October 3, 2003, at 07:53 AM, Justin Parke wrote:

> I am having trouble getting my "regular" (i.e. non "gifted and
> talented") physics students to understand how to determine the
> direction of a vector.  We have defined direction as the angle
> measured counter-clockwise from the +x-axis to the vector.
> 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.
>
> I have shown them how to construct a triangle from the vector
> components and the vector and then to use the angle to find the
> direction (i.e. add to 180 degrees or subtract from 360 etc).  Does
> anyone have success with something different?
>
> Thanks
>
> Justin Parke
> Oakland Mills High School
> Columbia, MD