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Re: vector components and notation



Joe Heafner wrote:
...
How can a vector *component* be both a vector and a scalar?

1) I agree that it is easy to find conflicting definitions
of "component". Both definitions are in wide use.

2) It isn't a life-and-death issue. If I ask somebody to
bring me the (scalar) element A_2 of the vector A, and
they bring me instead the (vector) A_2 x_2 where x_2 is
a unit vector, then my reaction is no-harm-no-foul.

3a) On the vanishingly-rare occasions where I really want
to specify the scalar version, I will ask for the "element"
as in "tensor element" or slightly sloppily the "matrix
element" since vectors are after all first-rank tensors,
and accordingly have elements.

3b) On the other hand of the same coin, if I really want
the vector version, I will ask for the _projection_ of the
vector A in the direction of some given vector x.

3c) You will notice that neither (3a) nor (3b) uses the
word "component" so the ambiguity is moot.

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

Representing vectors in terms of their elements is slightly
inelegant and should be avoided if possible. But often it
is not possible. For instance, if you want to represent
vectors on a computer, the usual representation is in terms
of elements in some chosen basis.

In contrast, any physically-significant result should be
independent of the choice of basis. So one ought to focus
attention on projections, such as the projection of one
physically-significant vector A in the direction of some
other physically-significant vector x.

The general expression for calculating the projection is:
Projection of A in x-direction = A.x
--- x
x.x

which can be simplified greatly using Clifford Algebra:
Projection of A in x-direction = A.x / x

since the reciprocal of x is written 1/x and is always
equal to x/x.x as you can verify by multiplying by x.

Note that A.x/x is, by the usual rules of precedence, to
be interpreted as (A.x)/x which is clearly not the same
as A.(x/x).

When forming the projection A.x/x we do not require that
x be a basis vector or even a unit vector. The result is
the same no matter what the length of x (assuming it's nonzero).


Bob LaMontagne wrote:

I'd like to thank all the posters who gave links
to either their own web sites or to sites that had introductions or
expositions of Geometric Algebra.

Here's another:
http://carol.wins.uva.nl/~leo/clifford/
especially the IEEE Computer Graphics articles (part 1 and part 2).

I think his asymmetric projection operator is less elegant than
the usual dot product, but it's important to see that Clifford
Algebra has applications other than mathematical physics. There
are a lot more jobs doing graphics and robotics than there are
doing mathematical physics. This provides a good way to motivate
students, to let them know that the stuff they learn in physics
class has real-world applications. Physics is not a closed
mutual-admiration society.