3D Vector with homogeneous component

Mar 2010
224
2
I know that a 3D Vector is represent with three coordinates. For example \(\displaystyle V = [3, 2, 5]\).

But I don't understand what a 3D vector with homogeneous component is. Does that extra component effect on the norm of a 3D vector? Or is the norm still \(\displaystyle | V | = \sqrt{3^2 + 2^2 + 5^2}\)?


Thank you for some explanation.
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey Nforce.

The definition of homogenous (like everything really) is different depending on who is defining it. In many mathematical contexts it refers to something involving a zero but I'm not sure what it means for your situation.

The norm of a vector should have the same definition for every vector in that space.

I am taking a look at this:

Homogeneous coordinates - Wikipedia, the free encyclopedia

and it says that homogenous co-ordinates refer to co-ordinates based on a projective space.

In a projective space, you don't have the same norm that you do in a Cartesian space: they are different kinds of spaces and subsequently they have different kinds of metrics, norms, and inner products (if these things exist for that space).
 
Mar 2010
224
2
So how can you calculate the norm then? Can you give me some example of a 3D vector with homogenous component and the norm for this 3d vector?
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Just to clarify though, are you talking about projective co-ordinate systems?