Quantifying the Physical Property of Direction.

[steveupson]

I can't even imagine describing a point in space without using some form of coordinate. At least not in any mathematics.

 

Please see my edit above.

 

The idea was to create a sort of protractor in order to handle direction directly (or independently). The length would then have become the vector dependent property.

 

I honestly don't know how I was expecting it to work, looking at it now. Although, if there were some sort of backdrop to set this whole thing on (which there isn't, is there?) then it might look more like what I thought it looked like until now.

Edited May 17, 2016 by steveupson

[Mordred]

True but I don't think the OP would distinquish the difference. It takes a good working knowledge to understand how dot products work.

Yes it's easy to apply the dot product between two vectors.

Hrrm I'm not sure if I can latex dot products lol. Here goes.

 

 

[latex]a \cdot b=|a|*|b|*cos(\theta)[/latex]

Hmm not quite right I don't want the dot on top of b but between a and b.

 

Ah there you use \cdot. (Thank god for online reference links lol. This time wiki lol)

 

Locally though manifolds resemble Euclidean space.

Edited May 17, 2016 by Mordred

[ajb]

The dot product of vectors is an example of an inner product, This is of course equivalent to the assignment of a metric. The dot product is then understood as coming from the Euclidean metric on R^n.

 

You can then use the metric to define the angle between two vectors, just as you do with the standard dot product. I am not sure how useful this notion is, I have never come across it.

[swansont]

 

Please see my edit above.

 

The idea was to create a sort of protractor in order to handle direction directly (or independently). The length would then have become the vector.

 

I honestly don't know how I was expecting it to work. Although, if there were some sort of backdrop to set this whole thing on (which there isn't, is there) then it might look more like what I thought it looked like until now.

 

 

I think that's the issue. You appear to be asking other people to explain how this could work, and the answer has been a consistent "it can't work". You say you can't do the math to show it, so why are you so insistent that this is viable?

[studiot]

The notion of direction has many interesting properties of its own.

 

Walk steadily away from the North Pole (ie place your back to it) and keep going.

 

Where will you end up?

Edited May 17, 2016 by studiot

[Strange]

That reminds me of the riddle: An explorer walks one mile due south, turns and walks one mile due east, turns again and walks one mile due north. He find himself back where he started. Where is he?

[Phi for All]

Walk steadily away from the North Pole (ie place your back to it) and keep going.

 

Where will you end up?

 

By my calculations, Starbucks.

[steveupson]

The dot product of vectors is an example of an inner product, This is of course equivalent to the assignment of a metric. The dot product is then understood as coming from the Euclidean metric on R^n.

 

You can then use the metric to define the angle between two vectors, just as you do with the standard dot product. I am not sure how useful this notion is, I have never come across it.

 

It doesn't seem to be useful for what I initially intended, although if it was paired with a reflection of itself it might do what I was thinking.

 

Or not. Needs more work.

 

 

I think that's the issue. You appear to be asking other people to explain how this could work, and the answer has been a consistent "it can't work". You say you can't do the math to show it, so why are you so insistent that this is viable?

 

Because my understanding of it is geometric and not algebraic. That's my best explanation. I can see where everything is pointing and it's all pointing differently than other objects that I've looked at.

 

I hope you understand that the directions (what I've been calling cardinal and ordinal) have a remote relationship with the function itself. They are not related to the function at all in the plane in which the two directions lie.

 

This is what made me think the angle between them could be quantified into a magnitude.

[Mordred]

Truthfully there comes a point where you need to attempt the math yourself. It's extremely difficult to develop a math for someone's else's idea when they themselves don't make any effort to post their own math.

 

For one fact there is no way to judge what your math skills are to develop a mathematical model that you may or may not understand.

Secondly it may help better explain what you are attempting.

[swansont]

 

Because my understanding of it is geometric and not algebraic. That's my best explanation. I can see where everything is pointing and it's all pointing differently than other objects that I've looked at.

 

I hope you understand that the directions (what I've been calling cardinal and ordinal) have a remote relationship with the function itself. They are not related to the function at all in the plane in which the two directions lie.

 

This is what made me think the angle between them could be quantified into a magnitude.

 

 

It can be: the angle between them. But that only provides a relative direction.

[studiot]

It should be noted that it is perfectly possible to separate the direction and magnitude attributes of vectors.

 

The zero vector has no direction (or indeterminate if you prefer)

 

Any vector in a direction field has no magnitude (or indeterminate magnitude if you prefer)

[steveupson]

Truthfully there comes a point where you need to attempt the math yourself. It's extremely difficult to develop a math for someone's else's idea when they themselves don't make any effort to post their own math.

 

For one fact there is no way to judge what your math skills are to develop a mathematical model that you may or may not understand.

Secondly it may help better explain what you are attempting.

 

I've always done the math myself. That's how I roll. I only look to see how other people do it when I either don't have time to figure it out for myself, or when it's too difficult for me to learn. Composing functions and rearranging algebra expressions is not anything that I've ever used. When you figure it out for yourself there's no real reason to write it down anywhere, is there? Other than to keep track of things during complicated operations. In those cases it's simple enough to invent the book keeping on the fly.

[Mordred]

 

Help me out with this, or at least let me know if anyone here can decipher what it is that Im trying to say.

 

Let the main axis of the new object lie along a line which is coaxial with a cardinal direction. Then, let the axis that is normal to the center of the small circle lie along a line which is coaxial with an ordinal direction. In the example used in the model, the directions of these two axes are oriented 45 degrees from one another in space. Also, the lines that are coincident with these axes lie in a plane, and are oriented at 45 degrees to one another in that plane.

 

The basic underlying principle behind the new technique is to express the relationship between these two directions as a function which incorporates all three Cartesian directions (x, y, and z), simultaneously. The new function can mathematically replace the vector (two dimensional) in describing the same (almost) direction information. The new function actually appears to operate with even more information which makes it communicative.

 

When the cardinal direction and the ordinal direction in our model are reversed, the same arithmetic relationship between the two will still exist.

 

The model that has been provided for a 45 degree relationship between two directions yields a different function from that in which this same model is used for defining a similar relationship between two directions that are not 45 degrees from one another. There is (should logically be) another additional function which will define the relationship between the 45 degree function and all of the other non-45 degree functions. An analogy would be how the trigonometric functions are used in plane geometry. A similar set of functions (as yet undefined) must exist for this geometry.

 

If someone could help me express this function of a function algebraically then I think everyone would have a much better opportunity to make some sense of my gibberish.

 

We shouldnt have to derive the actual function in order to understand how this works. But, once these functions are derived then it should be possible to express direction in units using this method.

 

This seems to be the gist of what your after but I'm still not clear on your understanding of vectors in particular your two dimensional section.

 

A vector in three dimensions can be expressed simply as

 

P_1=(x_1,y_2,z_3) with end point P_2(x_1,y_2,z_3)

 

Simply draw a line between those two points and there is your vector.

 

However I know this doesn't help for what your trying to do.

 

Probably because I don't particularly see the need for the above.

 

Let's explain we will use vectors for now. In 3D [latex]\mathbb{R}^3[/latex]

 

U is the set of vectors in [latex]\mathbb{R}^3[/latex] where[latex] (x,y,z)\in\mathbb{R}^3[/latex]

 

The four basis vectors being

 

[latex]e_1=(1,0,0).. e_2=( 0,1,0).. e_3=(0,0,1).. e_4=(0,0,0)[/latex]

 

Then using [latex]P(x,y,z),\acute{P}(\acute{x},\acute{y},\acute{x})[/latex]

 

Then [latex]u=(\acute{x}-x,\acute{y}-y,\acute{z}-z)[/latex]

Edited May 17, 2016 by Mordred

[steveupson]

You can then use the metric to define the angle between two *vectors* directions, just as you do with the standard dot product. I am not sure how useful this notion is, I have never come across it.

 

I'm not sure how useful it is either, but I think it's pretty cool.

 

This is a pictorial view of what I have been saying: file attached below

This seems to be the gist of what your after but I'm still not clear on your understanding of vectors in particular your two dimensional section.

 

 

 

This is my way of saying what you have also been saying. A vector must have a reference. The reference is always in a plane. All you are doing is triangulating by arranging three of these operations orthogonality.

When you look at the basic relationships that are determined by the geometry, it is clear that all of these relationships are rooted in plane geometry.

 

 

And I'm sorry, the rest of what you said is nothing like what I've been asking about, at all. I have a thread in the math forum that explains a new function. I've asked innumerable times for help expressing this function in algebraic terms.

 

I cannot make any more clear than that. It confounds me to no end that you still cannot hear what I am saying, at all.

 

Clue: whatever I'm trying to say algebraically should have a f() in it somewhere!

what the function returns.doc

Edited May 17, 2016 by steveupson

[Mordred]

Well this last document clears up a lot. What you have in the document is viable. I recall seeing something similar though I'll have to dig for it.

 

Though it's essentially a trigonometric application. I don't see why you want direction when your returning the magnitude of the vector. Though you can also calc each vector with the necessary details.

Ps this isn't anywhere close to what I interpreted your posts as asking for.

 

One key point though is this image allows to extrapolate 2d relations. I assume your looking for a 3d based on these relations with a different sphere reference.

 

 

Why would you think it requires a function???

 

https://www.mathsisfun.com/definitions/function.html

Edited May 17, 2016 by Mordred

[steveupson]

Why would you think it requires a function???

 

https://www.mathsisfun.com/definitions/function.html

 

Because it's a function.

 

Look at the animation: http://community.wolfram.com//c/portal/getImageAttachment?filename=NSTF.gif&userId=93385

 

The little numbers that are changing at the top of the image relate to the little numbers that are changing in the rectangular box.

 

The relationship is establish through a function.

 

 

 

Also, there is a function that is different, but similar to this for every direction.

 

 

Also, there is a function that will relate these other functions to one another. <---- This is what I want.

[Mordred]

Most trigonometric applications are functions but they are not

necessarily written as f(x).

 

For example an exponent function is

 

F(x)=b^x. So when see an exponent your looking at a type of function.

 

I think it's safe to assume much of your discussion on the math forum has been discussion on what constitutes a function. Particularly rereading Studiots post.

Edited May 17, 2016 by Mordred

[steveupson]

F(x)=b^x. So when see an exponent your looking at a type of function.

 

I think it's safe to assume much of your discussion on the math forum has been discussion on what constitutes a function. Particularly rereading Studiots post.

 

I'm with you so far, carry on.

 

(and no, just like what happened here, I haven't been able to convince anyone over there that this is a function, either)

 

 

It can be: the angle between them. But that only provides a relative direction.

 

 

After reexamining things, I do believe that there is a tether to orient things in the frame. It is the plane in which the two directions lie. The difference between this function and an angle is that angles don't commute and this function should commute.

It should be noted that it is perfectly possible to separate the direction and magnitude attributes of vectors.

 

The zero vector has no direction (or indeterminate if you prefer)

 

Any vector in a direction field has no magnitude (or indeterminate magnitude if you prefer)

 

We've come a long way since the days when I used to argue that this was a sphere. Thank you for dragging me out of the weeds on that.

[swansont]

After reexamining things, I do believe that there is a tether to orient things in the frame. It is the plane in which the two directions lie. The difference between this function and an angle is that angles don't commute and this function should commute.

Yes, two vectors define a plane, but there are an infinite number of orientations within a plane.

[steveupson]

Yes, two vectors define a plane, but there are an infinite number of orientations within a plane.

 

They aren't really unit vectors because those things relate to either a point or a surface.

Unit vectors can't be the metric but direction, once quantified, can be.

 

 

what the function returns 2.doc

what the function returns 2.doc

[steveupson]

 

By having an angle, you have a coordinate system. Otherwise, what is your angle relative to?

 

 

If alpha is a unique function of E then you only have one variable (E) and so you can only express directions in two dimensions.

In the same way that having a length does not necessitate a coordinate system is my guess. Representing a line without a coordinate system would seem awkward. We aren't really trying to represent an angle (analogous to a line), we're trying to represent the magnitude of the angle (analogous to a length).

 

 

I think that by using two projections it would be possible to measure the difference in direction between two lines that don't intersect in three dimensions. I'm not sure how one would go about it, exactly, but I don't see why it couldn't be doable, using math, of course.

[ajb]

If we have a Riemannian manifold then the angle between two vectors at a point can be defined, and this does not depend on the coordinates used. The expression is

 

[math]\cos \theta = \frac{\langle X, Y\rangle}{|X||Y|}[/math],

 

in hopefully clear notation. You can then pick some suitable local coordinates to calculate this angle.

 

Other than defining what we mean by orthogonal (or orthonormal) vectors (say for reducing the structure group of the tangent bundle), I have not seen this notion in use. Other than vectors being orthogonal I have not seen an application of this angle in differential geometry. Maybe there are, just I have not seen one.

[steveupson]

If we have a Riemannian manifold then the angle between two vectors at a point can be defined, and this does not depend on the coordinates used. The expression is

 

[math]\cos \theta = \frac{\langle X, Y\rangle}{|X||Y|}[/math],

 

in hopefully clear notation. You can then pick some suitable local coordinates to calculate this angle.

 

Other than defining what we mean by orthogonal (or orthonormal) vectors (say for reducing the structure group of the tangent bundle), I have not seen this notion in use. Other than vectors being orthogonal I have not seen an application of this angle in differential geometry. Maybe there are, just I have not seen one.

 

We are using the new function to replace [math]\cos [/math]. The difference is that we don't need a scalar length any longer. While this stuff does nothing to alter existing vector analysis, the two geometries cannot be readily mixed and matched. That's what threw me off track the other day.

 

​Every time a knowledgeable individual tells me "it won't work because of this," I try to look at everything in order to translate it into the new system. While trying to do this with circulation and swirl as studiot suggested, I sort of lost my footing there for a while, if you know what I mean. I'm still not sure if that stuff is translatable or not, it's way above my pay grade at this point.

 

The problem that everyone seems to be having is that they are used to looking at things a certain way. I certainly used to...

 

on edit> That isn't quite right, what I said about not needing the scalar length. what I should have said is that we don't use the length as the scalar or the metric. Length is the dependent part of the angle expressed by the inner product. Direction becomes the scalar.

Edited May 18, 2016 by steveupson

[studiot]

 

steve upson

I think that by using two projections it would be possible to measure the difference in direction between two lines that don't intersect in three dimensions. I'm not sure how one would go about it, exactly, but I don't see why it couldn't be doable, using math, of course.

 

 

These are called skew lines.

 

I won't offer you any more textbooks, these days on the internet there are many offerings on the subject.

Take your pick.

 

https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=angle+between+skew+lines&gbv=2&oq=angle+between+skew+lines&gs_l=heirloom-hp.1.0.0j0i22i30l3.922.6734.0.11750.24.19.0.5.5.0.203.2235.2j12j2.16.0....0...1ac.1.34.heirloom-hp..3.21.2420.bsVJXvT_MRc

 

steveupson

 

We've come a long way since the days when I used to argue that this was a sphere. Thank you for dragging me out of the weeds on that.

 

Every time a knowledgeable individual tells me "it won't work because of this," I try to look at everything in order to translate it into the new system. While trying to do this with circulation and swirl as studiot suggested, I sort of lost my footing there for a while, if you know what I mean. I'm still not sure if that stuff is translatable or not, it's way above my pay grade at this point.

 

 

I'm glad to see you are now trying to use the same terminology as every one else.

That will help your case no end.

 

As regards the circulation of a vector field, the maths is hard, but the explanation is easier.

 

If you consider any vector field and draw a closed loop in its space and take a unit (or other) vector for a walk around the loop then you can calculate an important property of that field called the circulation.

 

By take a vector for a walk I mean start somewhere on the loop (say point A) and align your moving (walking) vector with the vector field at each point of the loop until you come back to A (complete the loop).

As you move around the loop the walking vector will turn to point this way and that.

When you return to A it will be pointing in the same direction as when you started.

 

Now at each point of the loop there is another vector, the tangent vector of the loop.

Now form the dot product of these two vectors.

Now integrate (add up) all these dot products around the loop.

 

The thing is that for some vector fields, like gravity, this will be zero.

But for other vectors fields, like the velocity field of moving air, it may result in a calculable non zero number.

 

In Physics it is this circulation which is responsible for generating the lift force to allow aircraft to fly.

 

In symbols we have

 

[math]C = \oint {V \bullet dl} [/math]

 

 

Where C is the scalar circulation V is the field vector and dl is the differential length vector along the curve.

Edited May 18, 2016 by studiot

[steveupson]

 

These are called skew lines.

 

 

 

Thanks for this. I think that in the new system there won't be a requirement to translate the lines. instead there will be a requirement for a projection.

 

In Physics it is this circulation which is responsible for generating the lift force to allow aircraft to fly.

 

In symbols we have

 

[math]C = \oint {V \bullet dl} [/math]

 

 

Where C is the scalar circulation V is the vector and dl is the differential length vector along the curve.

 

 

 

Thanks for this explanation. I didn't understand that that was what they are doing.

 

 

 

 

I no longer feel a bit inadequate about my algebra skills. No one else seems to be able to do any better at composing the function.