Re: vector components and notation

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

Bob LaMontagne wrote:
> A minor quibble here - aren't projections the result of the dot product of the
> vector in question with the basis vectors - which makes them scalars (the A_x
> in your exposition)?

Taking a dot product is part of the story but not
the whole story when forming a projection.

In my book, the projections are vectors.  Specifically
the projection of vector A in a direction Q is a vector.

As previously discussed, the formula for calculating
the projection of A in the direction of q is
        A.q/q
or, if you are allergic to Clifford Algebra or
otherwise unable to multiply and divide by vectors,
the corresponding formula is
        A.q (q/q.q)

Note that in two dimensions, you can write A in
terms of its projections using the vector equation
        A = A.p/p + A.q/q
whenever p and q are orthogonal (not necessarily
orthonormal, and not necessarily "the" basis vectors;
only their orthogonality matters).

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

As to notation, in my book the expansion of a vector
A in terms of its matrix elements is written (in D=3)
        A = [A_1  A_2  A_3]
which is shorthand for
        A = A_1 x_1 + A_2 x_2 + A_3 x_3
where A_i is a scalar for all i, and x_i is the ith
basis vector.  Note that in the case of x_i the
subscript does _not_ indicate a component;  it just
indicates which basis vector.

If you are using boldface, x_i would use a boldface
x, since it is a vector.  But A_i is a scalar, and
should never be written boldface.

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

In the Clifford Algebra literature, it has become
standard notation to _not_ decorate vectors with
little arrows or boldface or anything else.  That's
because scalars, vectors, bivectors, etc. can all
be added and multiplied on more-or-less the same
footing.  That is, vectors are not sufficiently
special to require decoration.

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

Rick Tarara wrote:
> Understand that most of the quibbling over vectors, component vectors, basis
> vectors, projections of vectors, etc. is meaningless to 99% of our students.
> They are sufficiently confused over whether the x-component is Asin(theta)
> or Acos(theta).

That's all-too-true.  But that brings up a whole nother issue.
There are inconsistent conventions as to what "theta" means.
Engineering books are likely to do it differently from
math books.  Physics books are just inconsistent.  And the
conventions for D=2 polar coordinates conflict with conventions
for D=3 spherical-polar coordinates, often within the same book.

Fortunately, when you write the projection as A.x/x, the dot
product _always_ involves the cosine of the angle between A and x.

Any physical prediction should be independent of the choice
of such conventions.  The numerics of intermediate steps in
the calculations will depend on choices, but not the bottom-line
physical predictions.

>   It's of course best to be consistent and 'correct', but
> getting anal over it won't really help most students.

Right.  There is no 'correct' answer.  No global consistency
is possible.  All that is really required is to keep the
conventions consistent _within any particular problem_.