Re: vector notation

[John Denker <jsd@AV8N.COM>]

Aaron Titus wrote:
> ... applying this to intro physics. ... I don't know what I would do
> differently in an E&M course, for example.
....
> ... I started teaching students to use "direction cosines"
...
> However, I now ask students to always express a vector in terms of
> its components.

1) Good, that's a big step in the right direction.

2) Now you know how to handle the E&M course et cetera:  Primarily
   vectors and secondarily components of vectors.

In my experience doing physics, the occasions where components
are preferred outnumber the occasions where the magnitude&direction
representation is preferred by a ratio of 99:1 or more.

One reason for this is that finding a component (by projection)
is a _linear_ operation, whereas finding the mag&dir representation
is nonlinear.  Vector analysis is in some sense a branch of linear
algebra, and we use vectors [and tensors] because they are good
for representing physical systems that are linear [and multilinear].

Actually I'm beginning to think that my 99:1 number is an
understatement.  Although I've used the law of _sines_ on a
number of occasions, I don't recall ever getting any practical
use out of the law of _cosines_ in D=3.  Usually when I'm
dealing with angles in D=3, some other representation is
preferred, for example:
 -- A compound miter saw is built using Euler angles as its
  natural representation.
 -- If you need to reliably add one rotation to another in D=3,
  (e.g. autopilots or flight-sim games) you ought to be using
  quaternions aka Clifford algebra.
 -- In simple cases, pilots and engineers speak in terms of
  yaw, pitch, and roll ... which are Euler angles.
 -- Not to mention the occasions where you can forget about
  angles and just draw the vector from its components (coordinate
  of tail, coordinate of tip, connect the dots).


> now ask students to always express a vector in terms of
> its components.

I hope that "always" is an overstatment:
 -- Yes, virtually always, components are preferble to
  magnitude&direction, but
 -- No, not always-always preferable in absolute terms, in
  particular not preferable to doing without components
  whenever possible and just treating the vector as a VECTOR.

Any law of physics, indeed the answer to almost any practical
physics question, should be answerable in terms of vectors
_per se_, without any need to exhibit the components in any
particular basis (or even any need to choose a basis).  For
example, if the answer is F=ma, I don't want to see the
components of F or the components of a.  I just want to see
the expression F=ma as a relationship between vectors.  This
is not just a style issue; there are all sorts of practical
reasons as well as conceptual/pedagogical reasons for this,
which we can discuss if anybody is interested.

Vectors _per se_ are first-class geometric objects.  Vectors
need not be defined in terms of direction&magnitude nor in
terms of components.  Vectors can be defined "from scratch",
i.e. axiomatically, in terms of their vector properties.
These properties can be explored quite far without ever
mentioning magnitude&direction or components.  Indeed there
are situations (e.g. thermodynamics) where we have vectors
but no natural notion of magnitude or direction.

I know this axiomatic approach demands that the students
grasp a thorny concept:  They've spent their lives adding
and subtracting _numbers_, never dreaming that addition
and subtraction could apply to things that weren't numbers.
Well, deal with it.  Vectors are not numbers -- they're
vectors -- and we are going to add them and subtract them.