Re: vector notation

[Aaron Titus <titus@MAILAPS.ORG>]

First let me be explicit that I'm applying this to intro physics. I
haven't taught upper-level courses (we have no physics major), so I
don't know what I would do differently in an E&M course, for example.

Specifying direction using angles is ok for two-dimensional vectors,
but what about a three dimensional vector? A few years ago, I started
teaching students to use "direction cosines" , \alpha, \beta, \gamma or
\theta_x, \theta_y, \theta_z, which are the angles between the vector
and the given axis. These are easy to calculate since
\alpha=r_x/|\vec{r}| and so on.

I got very frustrated teaching "chapter 2" (or whatever chapter) on
vectors because students had so much trouble with their trig functions
and drawing vectors. Also, if they used atan to get an angle, they
really needed to also draw a picture since atan on the calculator
always returns an angle between -90 and 90 even if the angle is greater
than 90. I think that computer programs get around this problem by
giving atan arguments in the form of atan(y,x). Then, based on the sign
of each component, it returns the correct angle between -180 and 180.

I hoped that teaching direction cosines would serve two purposes: (1)
less confusion (i.e. they can learn the given equations and stick with
them; no other reasoning is necessary) and (2) some engineering
students will use such conventions in their engineering statics course.

However, I now ask students to always express a vector in terms of its
components. I use the convention of Matter & Interactions, namely

\vec{r}=<r_x,r_y,r_z>

We call it a "triple" and whenever I ask for a vector quantity,
students must always express it in this notation even if certain
components are zero.

On another note, I think we should do more with 3-D vectors. It helps
us to use a more "fundamental" approach than using an approach that
works well with 2-D vectors but not 3-D vectors. I think that as much
as possible, students should learn a single approach that works in most
cases. This will be especially beneficial for engineering majors, by
the way, who will work extensively with 3-D vectors. So why not teach
them an approach that will work both for 2-D and 3-D vectors?

AT


On Jul 22, 2004, at 9:19 AM, Robert Cohen wrote:

> The discussion on vector notation focused on using arrows
> or boldface for vectors.  I'd like to extend the discussion
> and ask what people think about the conventions for indicating
> the magnitude and direction of vectors.
>
> Regarding magnitude, in the past I've used |\vec{r}| as
> the magnitude.  However, the |\vec{r}| notation gets bulky,
> particularly when multiplying or dividing magnitudes.  For
> that reason, I'd prefer to simply use r (no boldface and no
> arrow).
>
> Which is more proper to use: |\vec{r}| or r?  Which is more
> useful from a pedagogical sense?  Does it matter if the vector
> is indicated by boldface vs. the arrow (or squiggly line)?
>
> Regarding direction, in the past I've used \theta_r as the
> direction of the vector \vec{r}.  However, using \theta_r
> for the direction makes it appear as though we are introducing
> a new variable rather than just extracting the direction
> information from the vector variable.  Besides, if the
> vector is \vec{F}_1, is the direction \theta_{F_1}?
> Too many subscripts for me and, I sense, for the students.
>
> Is there a notation for the direction that is similar in
> structure to that for the magnitude, like <\vec{r}>?  If not,
> would it be a problem to introduce an unconventional notation?
> I'm thinking of either using <\vec{r}> if I go with |\vec{r}|
> as magnitude or using \check{r} (r with a vee on top) if I
> go with r as magnitude.
>
> I use my own text so I have the flexibility to do whatever
> is best for the students (who, by the way, are not physics or
> engineering majors).
>
> P.S. Since this discussion started, I've encountered several
> websites that explicitly state that they use boldface because
> they are unable to reproduce the arrow on the website.
>
> ____________________________________________________
> Robert Cohen; 570-422-3428; www.esu.edu/~bbq
> East Stroudsburg University; E. Stroudsburg, PA 18301