What can I call the two possible directions on a line (as a category)?

In English, a vector is said to have two properties: a length and a direction. The possible directions correspond to half-lines out of the origin (so that, eg, up and down are different directions). In many other languages, a vector is said to have three properties: directions correspond to lines (so that a vector pointing up and one pointing down have the same up-down direction), and a third property determines which way the vector is pointing along that line (up or down in our example).

This may seem strange at first, but it's actually very useful to separate these concepts in mathematics. In fact, I need to do it in something I'm writing right now, but I can't find the right English words for it.

Firstly, I need a word to indicate that third property. Is there any accepted term for it? In the wikipedia article on vectors, there is a picture where it is labeled as "sense", although the term does not occur anywhere in the article itself.

Secondly, I might need a clear way to indicate which sense of the word "direction" I'm using (line or half-line). This does not need to be a single word, but it should be as clear and unambiguous as possible.

It would be even better if any official references for this usage could be found. This problem is bound to have come up before in the translation of foreign scientific literature.

Solution 1:

According to the Oxford English Dictionary, the wikipedia illustration is correct. Quoting the very last definition (29!) of "sense":

29.

a. [After French sens.] A direction in which motion takes place. rare.

b. Chiefly Math. That which distinguishes a pair of entities which differ only in that each is the reverse of the other.

The OED supplies some quotations; I've selected a few:

1894 H. W. L. Hime Outl. Quaternions i. i. 2 No two vectors are equal unless they have, first, equal lengths, and, secondly, similar directions—the phrase ‘similar directions’ meaning ‘parallel directions with the same sense’.

1947 R. Courant & H. E. Robbins What is Math.? (ed. 4) iii. 159 Although inversion preserves the magnitude of angles, it reverses their sense; i.e. if a ray through P sweeps out the angle x. in a counterclockwise direction, its image will sweep out angle y. in a clockwise direction.

1977 Holland & Treeby Vectors i. 10 The vector (1/a)a is a unit vector in the direction and sense of a.

Disambiguating "direction" is more difficult. Frankly I would suggest a translator's note at the beginning.

Solution 2:

I think a single word that describes it would be:

bearing

As in the bearings of the direction.

Solution 3:

By definition, a vector includes the direction and the sense is always away from the origin. In that case "up" would be something like (0,3) and "down" would be (0,-5) while left and right would be (-1,0) and (4,0) respectively.

The speed component of the vector is usually indicated by the "length" of the vector, so (0,4) is twice as fast as (0,2). Alternatively, a vector can be also be specified by an angle and a length. All of this translates to 3 or more dimensions.

So there's no real need for a third element. However, if you wanted to indicate going forwards or backwards against a vector I'd probably use "negative" and "positive".