Re: i,j,k things

["Glenn A. Carlson" <gcarlson@MAIL.WIN.ORG>]

> A coordinate system to represent F (a set of three axes) and a unit
> (to quantify forces) must be chosen to turn i,j,k into unit vectors.
> Without this they are only symbols.

Vectors (including unit vectors) are objects which exist independent of
a coordinate system.  And I don't mean this in the sense of Platonic
forms; rather, vectors are things (objects), abstract things perhaps,
which do not depend on what coordinate system we choose.  The coordinate
system only provides a representation, a name, for the vector.

E.g., A vector A may be represented in an unprimed coordinate system by
A = xi + yj + zk, but in a primed coordinate system by A = x'i' + y'j' +
z'k'.  It's the same vector, only the name we give the vector, its
representation, has changed.  A nonmathematical example is a person's
name:  I exist independent of my name.  I am the same person whether my
name is Bill in St. Louis, Billy in Atlanta, or Wilhelm in Berlin.

I tell my students that points and vectors in space exist independent of
the coordinate system we select.  We can select any coordinate system,
so we select the coordinate system most convenient to us.  E.g., for
inclined plane problems, we use a coordinate system normal and parallel
to the incline only because its the most convenient.  We could also do
inclined plane problems using a vertical-horizontal coordinate system,
but choose not to, only because it's more convenient use the
normal-parallel system.

I want my students to know from the start that they can solve Newtonian
mechanics problems using any coordinate system (any nonaccelerating
system) they like, but a smart choice of coordinate system will make the
problem easier than otherwise.  E.g., in free-fall problems, the student
can choose the origin to be at the ground or at the top of building or
anywhere in between; the positive direction can be chose as up or down.
It doesn't matter which is done; it's only a matter of ease and
convenience.

Regards.

Glenn A. Carlson
St. Charles County Community College
St. Peters, MO