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Re: ?negative vectors, ?positive vectors



I wrote:

To repeat: -- Vectors in R^3 never have signs. -- Vectors in R^D
never have signs, for any D>1.

Then at 05:28 PM 9/11/01 -0400, James McLean wrote:

I'm surprised by the exception of D=1. Should a "vector" in D=1 be
treated any differently?

It does not NEED to be treated differently,
but you MAY treat it differently if you wish.

In D=1, things can be well-ordered. For larger D they cannot. Typical
vector operations do not require things to be well-ordered, so this little
fact doesn't usually matter much.

If you want to start with vectors and add a concept of well-ordering, you
must restrict yourself to D=1 real vectors. (Note that complex numbers are
not well-ordered, even in D=1.)

If you wish, in D=1, you might choose to only
discuss the components.

Yes. And that's not even restricted to D=1; even in higher dimensions you
can discuss vectors in terms of their components. But in some sense the
vector has a physical reality that transcends its representation by
components.

a vector still only has direction and magnitude.

Right.