I recall learning the phrase "a vector is something with magnitude and direction" many years ago. It dawned on me that "magnitudes" are well defined as numbers (or numbers with units), but I have never heard of a definition of a "direction." I know unit vectors can be treated like directions, but they are still technically vectors. I figure a direction would have the following properties:
- A direction times a number is a vector
- The inner product of a direction with a direction is a number
- Directions cannot be added or subtracted
The motivation is to mathematically define directions like "north" or "east." It makes sense to say "4 meters north" or "6 meters east," and makes sense to say "4 meters north and 6 meters east" but it makes no sense to say "north and east" because you never specified how much north or east. However, it is perfectly reasonable to say $north\cdot east=0$ because north and east are perpendicular.
One could go further and define outer or exterior products between directions and inner products between the resulting entities.
So it seems to me there is decent motivation to formalize the notion of a direction, but I am unaware of whether anyone has done it.