How to encode direction information?

[mnb96]

Hello,
I was wondering how I could express a geometric quantity which has only magnitude and direction but no orientation (sense)? and perform ordinary operation on that.

For example:
it is clear that a vector is not appropriate because vectors with same magnitude-direction but opposite orientation will cancel out: instead I want that two entities with same magnitude and direction will result in another entity with the same direction and double magnitude.

Is this possible to do?
I've heard somewhere that this is probably done using tensors, but I have no idea how. Maybe I misunderstood.
Any idea?

 

[Tac-Tics]

You can replace the idea of a vector's direction with a line passing through the origin. This eliminates the distinction between positive and negative lengths, since, unlike a vector, a line isn't "pointing" either direction. On top of that, you would simply need to pair it with a non-negative scalar to achieve what you seem to be looking for.

However, note the properties you must conclude such a system of objects has. Since magnitudes can only be positive, adding two of these objects will always produce a third with equal or greater magnitude. There are no additive inverses. There is no scalar multiplication by negative reals. These things don't have natural definitions.

Essentially, such a definition is what you get by removing all vectors in half of the plane (in R^2). I don't think such a system of objects would be terribly useful. What applications do you have in mind?

 

[mnb96]

Perhaps I formulate my question ina misleading way. I think the quantities I am looking for, are extremely useful and are apparently very used in physics.
I finally found something related:

http://www.cs.auckland.ac.nz/~burkhard/PhD/img13.html
http://www.cs.utah.edu/~gk/papers/tvcg00/img144.png

Maybe that gives a better idea of what I am after.
As you can notice, it is possible to construct such entities that resembles vector fields but instead of vectors they have ellipsoids (not straight lines),

Any hint about this topic or useful link is always appreciated! As I don't understand what could be a good starting point for learing how to manipulate these things.