## Thursday, 2 October 2008

### Directions in 3D space are 2D

In something of a response to one of the whinges about my previous post (that all vectors are a direction and a magnitude), I hereby declare that all directions in 3D space are in fact 2D, therefore do not count as 3D vectors.

Eh?

okay, imagine any vector for a direction in 3D space, you can map it to a sphere (remember, directions are unit vectors, so there's no 0,0,0, nor any vector of any length other than 1), and as you know, places on this world can be pinpointed on a map.

... yes? more?

if you don't need anything other than 2 dimensions to fully describe a vector of values, then that's how many dimensions it's got. Any production of more values from the lower order vector of values is just a function of them.

consider this, if you were told that foward acceleration was a vector, you could say that it was not true, in fact it was a 1 dimensional vector (scalar), because any acceleration that wasn't forward didn't classify as a forward acceleration, therefore you only need one value to descibe the acceleration.

3D -> 1D

Care for more?

How about the horizon? point to somewhere on the horizon... you can give me a value, a single value, back, and i can know where you were pointing. That's right, up and down, left and right, and how far away the thing you were pointing at can be reduced down to one number.