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Re: vector components and notation



John,

I'm on the same page as you now and understand what you are describing. The
price of the notation I advocated is always using |A| absolute value bars to
indicate magnitude or lengths of vectors for internal consistancy. I think
I will want to rethink how I teach this in the future.

I must confess, that in practice I am not "anal" about the use of the
absolute value bars and tell my students that context often makes it clear
which is being meant. I don't think I'm confusing my students any worse
than when I have used the approach you and H&R use. I have imagined that
using the H&R conventions lead to some confusion (or lack of recognition)
that a_x <bold/>i_hat</bold> is simply a vector though a very special one.

This could be an interesting PER paper I suppose, to see which notational
conventions cause more confusion.

Joel Rauber

P.S. What are the opinions of people on this list regarding Algebra Based
books that do not introduce the unit vectors i,j,k ?

-----Original Message-----
From: John Mallinckrodt [mailto:ajm@CSUPOMONA.EDU]
Sent: Wednesday, September 18, 2002 9:25 AM
To: PHYS-L@lists.nau.edu
Subject: Re: vector components and notation


Joel Rauber wrote:

John Mallinkrodt wrote:

> If there is a vector indicated by <bold>A</bold>_sub_x, then
> <italic>A</italic>_sub_x should formally represent the
*magnitude* of
> that vector and, thus, would never have a negative value
as it could
> if it instead represented the scalar x-component of the vector
> <bold>A</bold>.

I'm not sure I follow:

Agreed. See below.

let me restate and clarify the notation that I
believe is consistant

(Small note: I'm a proponent of arrows over letters for
vector quantities
instead of boldfacing. But I'll follow what you have done
above for ASCII
clarity.)

Ditto

<bold>A</bold> = a vector

<bold>A_x</bold> = a component vector, John D's projection onto
<bold>i_hat</bold>

A_x = a scalar component = <bold>A</bold> dot <bold>i_hat</bold>

|A_x| = a positive scalar, the length of the component vector
<bold>A_x</bold>

It does appear that I am still failing to make my point.
I'll try once more:

By convention we write

|<bold>A</bold>| = magnitude of the vector <bold>A</bold>

but we generally allow ourselves the notational shortcut

A = |<bold>A</bold>|

In other words, when a textbook writes a quantity in bold face, it is
understood to be a vector and when that *same quantity* is written in
regular (but usually italicized) face, it is understood to represent
the *magnitude* of the vector which is always *nonnegative* quantity.
Thus, if we allow ourselves to refer to the vector

<bold>A_x</bold>

then, logically, the quantity

|<bold>A_x</bold>| = A_x

represents the *magnitude* of *that* vector and shouldn't *also* be
used to represent the scalar x_component of the (different) vector

<bold>A</bold>

To avoid this ambiguity, I simply prefer never to use the notation

<bold>A_x</bold>

John
--
A. John Mallinckrodt http://www.csupomona.edu/~ajm
Professor of Physics mailto:ajm@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223