Are Vectors Always Positive in a 1-Dimensional System?

[Born2bwire]

Another example, in polar coordinate system 3 cos \theta

That is not a vector, that is a scalar function of theta.

In polar coordinate system i.e

I've seemed to forget vectors completely...

You misunderstand what a vector is in polar/cylindrical coordinates. A vector in polar coordinates is not defined by a magnitude and an angle, that would only give you the subset of vectors that originate at the origin. Let's say I have a vector from the point (1,1) to (0,1). Your definition of magnitude and angle would not be able to represent this vector using the polar/cylindrical coordinate basis vectors. In cartesian coordinates the vector is (-1,0) from the point (1,1). In polar coordinates, the vector is (-1/\sqrt{2},1/\sqrt{2}) from the point (\sqrt{2},\pi/4). The location of points in polar coordinates is defined by magnitude and angle, but not vectors.

A general vector is made of a magnitude and a direction and it usually has a location. The magnitude and direction are represented as a combination of orthonormal bases that span the coordinate space. This way we can use vector operations to manipulate the vectors in such a way that will always give us a vector in our coordinate space without a lot of calculations. The use of the basis vectors makes the vector operations much simpler. Think how you would have to add the vectors, in Cartesian coordinates, (1,0) and (0,1). If you just described them mathematically as a magnitude and angle, then you have to do the conversions to see how their directions add. Now displace these vectors off of the origin and the conversions become more complicated.

 

[tiny-tim]

are you saying that you think a vector v is defined by a magnitude (|v|) and an angle?

In polar coordinate system i.e

I've seemed to forget vectors completely...

Yes, I remember you said …

What do you mean by (a,b)...those are coordinates right?

dE_logics, the most important thing about vectors is that they "add like vectors" …

you can use the parallelogram rule, or just add the Cartesian (ie, x y z) coordinates …

but the thing that distinguishes a vector from a scalar or a spinor (maybe you haven't come across them yet) is the way they add.

Now, adding polar coordinates is really difficult (eg what is (r,θ) + (s,φ)?), and almost impossible if they don't start from the same point.

But adding Cartesian coordinates is really easy … (x,y,z) + (a,b,c) = (x+a,y+b,z+c) … even if they don't start from the same point.

(Same for cross product and dot product.)

Polar coordinates are a useful calculation tool in some physical situations (usually where there's a "central" force), but usually they're a nuisance (and that's in 2D … in 3D it's even worse ).

If you've "forgotten vectors completely" and are trying to re-learn them, then concentrate on Cartesian (ie x y z) coordinates.

 

[Klockan3]

That is 100% incorrect.The additive inverse operator is an operation that can applied to a vector. It is not an intrinsic part of the vector. Vectors have a magnitude and a direction. Those are intrinsic parts of the vector. Rather than repeating what HallsofIvy said, I'll simply quote him:

You can order vectors, read some order theory... Also that was a one dimensional vector, which can be ordered really easily so then they absolutely have signs even with the most layman use of the word.

Anyway, that have nothing to do with this thread at all, the numbers inside vectors have signs and as such you could say that the vector as a whole have "signs".Anyhow, the most important question in this topic is:
dE_logics, how much maths have you read? To me it sounds like you haven't even read any linear algebra since you haven't seen the standard notations for vectors etc. This is really important so that we can understand what problem you might have.

 

[D H]

You can order vectors, read some order theory...

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.

Anyway, that have nothing to do with this thread at all, the numbers inside vectors have signs and as such you could say that the vector as a whole have "signs".

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

 

[dE_logics]

That is not a vector, that is a scalar function of theta.



You misunderstand what a vector is in polar/cylindrical coordinates. A vector in polar coordinates is not defined by a magnitude and an angle, that would only give you the subset of vectors that originate at the origin. Let's say I have a vector from the point (1,1) to (0,1). Your definition of magnitude and angle would not be able to represent this vector using the polar/cylindrical coordinate basis vectors. In cartesian coordinates the vector is (-1,0) from the point (1,1). In polar coordinates, the vector is (-1/\sqrt{2},1/\sqrt{2}) from the point (\sqrt{2},\pi/4). The location of points in polar coordinates is defined by magnitude and angle, but not vectors.

A general vector is made of a magnitude and a direction and it usually has a location. The magnitude and direction are represented as a combination of orthonormal bases that span the coordinate space. This way we can use vector operations to manipulate the vectors in such a way that will always give us a vector in our coordinate space without a lot of calculations. The use of the basis vectors makes the vector operations much simpler. Think how you would have to add the vectors, in Cartesian coordinates, (1,0) and (0,1). If you just described them mathematically as a magnitude and angle, then you have to do the conversions to see how their directions add. Now displace these vectors off of the origin and the conversions become more complicated.

Yeah, using solely polar convention, we won't know the direction at which the vector 'points'.

Its like a line segment in a Cartesian coordinate system.

In polar coordinates, the vector is (-1/\sqrt{2},1/\sqrt{2}) from the point (\sqrt{2},\pi/4).

aaaa...sorta didn't get that, never mind

dE_logics, the most important thing about vectors is that they "add like vectors"

i.e algebraic addition is different...right?

(maybe you haven't come across them yet)

You're right!

Polar coordinates are a useful calculation tool in some physical situations

Yeah I did lots of questions on that (though not without understanding)...phaser diagrams are an example, in representing complex numbers etc...

 

[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.