Re: non-vectors

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

Joe Heafner wrote:
> Besides finite rotations, what other quantities have both magnitude and
> direction but are not vectors? I suppose we could include pseudovectors
> here, but I'm looking for something simpler.

This is a spectacularly good question.

A good answer requires slightly restating the question,
but that's not a problem.  Here goes:

1) The name vector is applied to objects that exhibit certain
invariances with respect to certain transformations.
 -- The magnitude of a vector is invariant under rotation.
 -- Ditto for reflection.
 -- The direction of one vector relative to another is
    invariant under rotation.
 -- Ditto for reflection.

2) The name pseudovector is applied to objects that exhibit
certain invariances with respect to certain transformations.
Pseudovectors behave the same under rotation but change sign
under reflection.

Vectors (and pseudovectors) in D=3 space can be represented
by an array of three numbers.  The converse is not true; not
every array of three numbers is a vector.  It could just be
a shopping list, with no particular transformation properties.

A rotation is _not_ a vector.  You can specify a rotation in
terms of three numbers such as the Euler angles (yaw, pitch,
and roll) but these do _not_ behave properly when you try to
"add" rotations or "rotate" rotations.  And you are at risk
because of a singularity in the coordinates ("gimbal lock")
which occurs when the pitch angle is 90 degrees.

3) The name spinor is applied to some other objects that
exhibit interesting transformation properties.  You can
represent rotation space in terms of spinors;  you get a
double covering of the space, but at least it is nonsingular
(no gimbal lock).  This representation is highly recommended
if you are writing a flight simulator or flight-control
system.  The term "quaternion" is more-or-less synonymous.
These things can be represented in terms of Pauli spin
matrices.

In some sense a spinor is the square root of a vector; it
takes two spinors to make a vector.

4) Moving up the totem pole now, what do you call something
that transforms like two vectors?  Answer:  A tensor.
Example:  Consider the tensor of inertia of an irregular
object such as a potato.  The moment of inertia clearly
has magnitude, and it transforms under rotations in a very
nice well-behaved way, but it is not a vector.  It is a
second-rank tensor.

5) There exist higher-rank tensors.  You don't run into
such things too often in introductory physics classes, but
they certainly exist.  One example:  suppose you set your
gyrocompass to "north" at one location, and then move to
another location on the earth.  How much does it disagree
with the local "north" at the new point?  It depends on
a vector describing how far you have moved, another vector
describing the curvature of the earth in various directions,
and there will be a third vector that describes the effect
on the gyroscope.  So there must be a third-rank tensor
involved.

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

To summarize:  Things we know about with definite physical
significance and well-behaved transformation properties
include:
 -- spinors
 -- first-rank tensors i.e. proper vectors and pseudovectors
 -- second-rank tensors
 -- et cetera.