[no subject]

["John S. Denker" <jsd@MONMOUTH.COM>]

At 01:23 PM 9/10/01 -0500, Tina Fanetti wrote:
> I have a problem with my calc-based physics students.
> The problem is over the sign of g, the gravitational acceleration.

Part of the problem is that the vector g doesn't have a sign!!!!!

A vector has a _direction_ and a _magnitude_.

Saying a vector is "less than zero" cannot possibly mean anything.  The
less-than operator is not defined for vectors.  Consider the contrast:
 a) Points on the line are well-ordered, so the less-than operator means
something.
 b) Points in the plane are not well-ordered.  Points in D=3 space are not
well-ordered.  That is, the less-than operator is meaningless and cannot be
made meaningful

To repeat:
 -- Vectors in R^3 never have signs.
 -- Vectors in R^D never have signs, for any D>1.

======================

For the vector g (bolddface), its magnitude |g| is positive.  The magnitude
of _anything_ is positive.  It would be completely wrong to write |g| =
-9.8 m/s^2 or anything like that.

A vector can have components.  The numerical value of each component
depends on what the vector is AND on what coordinate system we have chosen.
 The choice of coordinate system is arbitrary, and any quantity of physical
significance will be independent of the choice of coordinates.  (Yes, this
means that the numerical value of a coordinate has no physical significance.)

In the conventional coordinate system where the Z axis points vertically
upward, the z-component of the g vector will be a negative number -- but
this is not something of physical signficance;  it depends on the arbitrary
choice of Z axis.

a) It is 100% OK to say that the vector g has a downward direction.

b) It is !not! OK to express this by writing
        g (boldface) = -9.80 m/s^2                      !??!
   That equation is dimensionally unsound;  it equates a vector
   to a scalar.

c) It is OK to say g dot Z is negative, for certain arbitrary
   choices of Z.  The result of a dot product is a scalar.
   Scalars are well-ordered.

> I have told them the convention is that it is always
> downward even if the object is going upward.

Yes, that's the correct way to say it: the g vector has a direction that is
always downward.

> I can't seem to explain to them why this is.
> They tell me the book says it is positive.

It depends on what book, and on what "it" refers to.
The symbol g (not boldface) in many standard systems of notation
is used to represent the magnitude of the vector g (boldface) and
in that case g (not boldface) is definitely a positive number.

> How can I make it clearer to them about g and its sign?

Don't say that.  It's not correct to say it that way.  It is not true that
g has a negative sign.
 -- The symbol g (boldface) is a vector, which has a direction, but that
direction is not properly expressed by a sign, or by an equation of the form
        g = - something
 -- The symbol g (not boldface) is a scalar magnitude, and it is most
certainly positive.