Re: speed and velocity

[Phil Parker <PPARKER@TWSUVM.UC.TWSU.EDU>]

> Date:    Tue, 1 Dec 1998 20:29:18 +1100
> From:    Brian McInnes <bmcinnes@PNC.COM.AU>
>
> > Date: Tue, 1 Dec 1998 4:02 AM
> > From: Phil Parker <PPARKER@TWSUVM.UC.TWSU.EDU>
>
> > In mathematics we distinguish between scalar components and vector
> > components for precisely this reason.  We would say Bob
> mentioned vector
> > components, and you've described scalar components.
> >   Also in mathematics, vectors can have signs, but the
> sign attached to a
> > vector is conceptually different from the sign attached to
> a number. Using
> > the same minus symbol for both is a bad idea from a purely
> conceptual view,
> > but it's very human.
>
> Phil,
> Let's see if I understand you.
> (1) By "vector components" do you you mean scalar components
> of a vector multiplied by a unit vector along the
> appropriate axis?
Depending on what you mean by "axis" the answer might be "yes."
Resolving acceleration into tangential and normal "components"
is an example of what we call vector components. For a plane
trajectory, the relevant axes would be the tangent and normal
lines.

> (2) Why do you want (or need) signs for vectors?
Ugh. From a pure, abstract, mathematical viewpoint, we *define*
vectors as things that can be "added" and "subtracted" (among
other properties, which aren't relevant yet).  We also insist
that subtraction is a shorthand for adding an "additive inverse"
so that  u - v  "really" means  u + (-v) .  (If you know the
jargon, it's an abelian group; if not, try to pretend this
sentence isn't here.)  Then it requires a proof that -v is the
same thing as the scalar multiple (-1)v.  (I warned you this
was "pure math".  I'm not sure how relevant to physics this
view is.)  We then interpret the proof as telling us that we
may happily confuse the two "minus signs" -- whenever we can
get away with it.  But we have to remember that they're really
different whenever it matters, which fortunately isn't often.

> In replying to Bob Sciamanda
Actually it was to Dewey Dykstra.

> you say "It is important to
> keep the distinction between numbers as numbers (scalars)
> and numbers as 1-dimensional vectors.  This post thoroughly
> confuses the two senses."  What I find confusing is the idea
> that numbers, which are pure magnitude (surely) can be
> vectors (1-dimensional or otherwise)!
You're right: they aren't.  BUT ... I think almost everyone uses
the real number line to represent 1-dimensional vectors via this
geometrical method:  the number 1 is identified with the vector
(thought of as a directed segment) from 0 to 1, etc.  So then we
have real numbers centered on 0: positive numbers representing
right-pointing vectors and negative numbers those left-pointing.
We also have real numbers lined up from left to right as usual:
scalars.  So if you think -2 < 1 then you think of -2 and 1
as purely real numbers: -2 is to the left of 1.  But if you think
"-2 < 1 doesn't make sense because -2 is farther away from 0 than
1 is"  then you're thinking of -2 and 1 as vectors: directed line
segments from 0 (tail at 0, head wherever), you're comparing
them by comparing lengths (positive numbers), and 2 > 1. (In math,
we wouldn't use "magnitude" for a negative number, only for
positive ones and 0.)  Mathematicians say < and > aren't
defined for vectors, only for scalars (numbers).
   If you don't like calling them numbers for this, call them
points on a real line or whatever.  Sometimes we think of those
points as numbers, sometimes we think of those points as vectors,
and mixing the two always leads to trouble.  (What does "smaller"
mean?  Applied math usually agrees it means "closer to 0, on
either side."  But in pure math it can mean "more left." Sigh.)
   I wish I could draw pictures and point; I think it would make
all this much easier to follow.
   I think the moral of the story is: vectors in 2 or more
dimensions are easier to understand because we use a special
notation for them ab initio; vectors in 1 dimension are confusing
because we don't see the need for a special notation until it's
too late.  (And when we get it figured out, we don't need the
special notation -- and everybody gets it figured out well
enough to cope eventually, so why bother?  Believe it or not,
the New Math of the '60s came largely from a desire to eliminate
such sources of confusion -- and there's still fallout from that
disaster.)

---------------------------------------------
Phil Parker        pparker@twsuvm.uc.twsu.edu
Random quote for this second:
  It's easier to get forgiveness for being wrong than
  forgiveness for being right.