Are Vectors Always Positive in a 1-Dimensional System?

[dE_logics]

dE_logics, how much maths have you read?

XII grade.

The course is pretty higher end here, i.e involving advanced calculus and all (its engineering mathematics actually)...but no one understands what's happening, they just know how to 'solve' the questions.


I'm the only student...or actually person in general (including most of the teachers) who tries to atleast grasp the concept...which most teachers can't explain.

 

[dE_logics]

Vector is different from algebra right?...they are different topics.

 

[tiny-tim]

vector spaces

Vector is different from algebra right?...they are different topics.

Vector spaces are part of algebra …

a vector space has a scalar field, a dimension, and a (coordinate) basis …

see http://en.wikipedia.org/wiki/Vector_space for details

 

[dE_logics]

Ok, let's just forget about this, I'm redoing this.

 

[Klockan3]

I suggest you follow your own advice. The best one can do in general with vector spaces is to impose a quasiorder. You cannot impose an ordering (i.e., a total order) on vectors of dimension 2 or higher.

Of course you can, you can even make a bijection between R and R^n without much trouble which means that you can create an isomorphism between them which in turn means that if we can order R we can also order R^n. They have the same cardinality. Read some more, you can make a total order on everything that have less than an infinite amount of dimensions at least.

What is the sign of $${\boldsymbol x} = \hat {\boldsymbol i} - \hat {\boldsymbol j}$$?

It depends on how you define them. There is no single ordering of vectors.

Anyway, as I said this is way beyond what the OP is discussing, he just wants to know what a minus vector means and I already said that it is the minus operator which mirrors stuff in the origin which is operating on the vector and that in reality the whole notion of sign is a construct just to make it easier to understand elementary maths which means that anything rigorous can't be said about it without defining it further than how you do it in elementary.

Like, they say that permutations have signs even though it have nothing to do with the kind of signs discussed in the "normal" maths.

 

[dE_logics]

Since we have big people discussing on this, I think the actual answer unknown for most.

 

[JasonRox]

? A "scalar" is a member of the field the vector space is defined over. In many applications, the field is the field of rational numbers or field of real numbers. And those certainly do have signs!

To me it just comes down to a "sign" is just a more convenient to write the additive negation to an element. If we want, we can say the additive negation to the vector B is $B_i$ regardless of field. (Obviously, this is not the best way to go.)

To me this question has really no meaning.

Note: It's been so long that I ever thought of vectors over the real numbers. I almost completely forgot. I was always thinking about it over a field, and yes usually the rational numbers. But most books teach it generally, and then apply a specific field.

 

[dE_logics]

Vectors do not have signs except in a 1-d system.

Also the cannot be represented by just 1 number in a 2-d or more system.