4D Angles?

[Mukilab]

Hello, I recently got informed of how 3D 'angles' work, e.g. in a sphere type 3D 'radius'.

 

However, I was thinking of how one would even think of a 3D angle? Or is there already a measurement for one? If one took a sphere and made it 4D would a 4D angle look as if a sphere in a sphere or a torus? (ironic because this is how the universe is hypothesized to look like )

 

Aplogies if the last sentence is irreadable (unreadable?), I don't know how to describe something if I don't know what it is

[Bignose]

Another recent thread talked about this to a certain extent. The inner product or dot product between two vectors in 2 or 3 dimensions can have a be thought to determine the angle between them.

 

[math]\mathbf{u}\cdot\mathbf{v}= \sum_i u_i v_i = \lVert\mathbf{u}\rVert \lVert\mathbf{v}\rVert \cos \theta[/math]

 

The exact same thing can be done for any i>1, however. I.e. you can use i=4 in the same equation, and find a [math]\cos\theta[/math] that makes the equation true. That can be defined as the cosine of the angle between the 4-vectors.

[Mukilab]

sorry, is there any way to verse that in a simpler form? I'm rather a 'simpleton' and I got as far as the the first paragraph.

 

P.S. does this make sense? "can have a be thought", if it does, what is a "be thought"?

[the tree]

sorry, is there any way to verse that in a simpler form? I'm rather a 'simpleton' and I got as far as the the first paragraph.

Not really.

 

For two n-dimensional vectors: [imath]\mathbf{u} = ( u_1 , u _2 , u_3 , \dots , u_n)[/imath] and [imath]\mathbf{v} = (v_1 , v_2 , v_3 , \dots , v_n)[/imath], their "dot product" is defined as:

 

[math]\mathbf{u}\cdot\mathbf{v} = \sum_{i=1}^n u_{i}v_{i} = u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}\dots+u_{n}v_{n}[/math]

 

The length of a vector is defined by it's "norm":

 

[math]\left\|\mathbf{u}\right\| = \sqrt{\mathbf{u}\cdot\mathbf{u}} = \sqrt{u_1^2 + u _2^2 + u_3^2 + \dots + u_n^2 }[/math]

 

And the cosine of the angle between two vectors is defined as:

 

[math]\cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{\left\|\mathbf{u}\right\| \left\|\mathbf{v}\right\|}[/math]

 

Are there any terms that you particular don't understand there?

 

If you want to think of it more intuitively, then any two vectors are co-planar. That is, there exists a flat plane which they lie on and the angle between those two vectors can be seen on that plane in the 2-dimensional sense that you're used to.

 

if it does, what is a "be thought"?

Almost surely a typo. Edited May 13, 2010 by the tree

[D H]

Back up a second, everyone. Bignose and the tree, you two are talking about the inner product. Mukilab appears to be talking about an extension of solid angle to higher dimensions. Those are completely different things.

 

Edit

Here's an article on computing solid angle in higher dimensions: http://karthikshekhar.wordpress.com/2009/05/23/computing-the-solid-angle-for-a-d-dimensional-sphere/.

Edited May 13, 2010 by D H
Addendum

[Mukilab]

Back up a second, everyone. Bignose and the tree, you two are talking about the inner product. Mukilab appears to be talking about an extension of solid angle to higher dimensions. Those are completely different things.

 

Edit

Here's an article on computing solid angle in higher dimensions: http://karthikshekhar.wordpress.com/2009/05/23/computing-the-solid-angle-for-a-d-dimensional-sphere/.

 

Thank you, the other posts were as clear as mud to me.

 

P.S. I'm not playing on the double irony

P.P.S If you don't understand the P.S then I am referring to the scientist called Mud whos theory/hypothesis (I can't remember what it was) was thought to be incorrect by every respected member of the scientific community earning the term 'as clear as mud' however he was later proved in fact correct, hence irony and double irony.

[D H]

To be brutally honest, if you don't understand the concept of an inner product then higher dimensional geometry really is not something you should be looking into yet. You need to learn to walk before you can learn to run ...

[Mukilab]

agreed but if I am not able but I can learn to walk, then I have to run.

[Mr Skeptic]

So long as there is only one angle, then as the tree said, you just redefine your coordinate system so that the angle occurs in a plane, then it is just a multidimensional angle that looks no different than any 2D angle. Since you need 3 points to define an angle and 3 points define a plane, you will always be able to do this. Well, if your space is flat anyways.