What is the 'sense' of a vector? In my country we are taught of vectors as if they have three components, module (the length), direction (slope of the line that contains the vector), and 'sense' (sentido), which indicates the "way" or "sense" that the vector "goes". The vector $\vec{u}(5,6)$ has the same sense as $\vec{v}(10,12)$, and the opposite of $\vec{w}(-5,-6)$. My question is: What is really the thing that we're talking about? Is it a number? What are the values that 'sense' can be? Is it either 'the same', 'opposite', or 'different'? What is the sense of $\vec{z}(5,4)$ compared to the first vector? Does it only exist while we're comparing vectors?
 A: The best mathematical translation of the Spanish sentido is direction rather than sense. 
It makes sense to say that two collinear vectors have the same or have opposite directions, but, absent some additional external reference, it does not make sense to speak of the direction of a vector or to say that two noncollinear vectors have or have not the same direction.
Given a vector, $(x, y)$, one can regard the unit vector $\tfrac{(x, y)}{\sqrt{x^{2} + y^{2}}}$ having the same direction as $(x, y)$ as its direction vector (or its sense). 
A: Sense is a set of all half lines who have the same orientation and by pairs belong to the same part of the half plane they define.

What is really the thing that we're talking about?

It is a set of half lines. A vector has sense A if the half line taken by extending the line segment from the end belongs to A.

Is it a number? 

No.

What are the values that 'sense' can be?

This makes no sense (haha) if we define it as a set.

Is it either 'the same', 'opposite', or 'different'?

You need to rephrase those in terms of set relations and/or new relations you define.

What is the sense of ⃗ (5,4) compared to the first vector?

A disjoint set and whatever else you want to define.

Does it only exist while we're comparing vectors?

No.
PS: I found this topic really interesting since I was taught the same thing and had a similar question.
