What is Vector?

Given two points $A$ and $B$, the vector $\vect{AB}$ represents the displacement from $A$ to $B$: it has a direction (the direction of the line $(AB)$, together with the way of traveling along it, from $A$ towards $B$) and a length (the distance $AB$, also written $\norm{\vect{AB}}$). Two vectors are equal when they encode the same displacement: $\vect{AB} = \vect{CD}$ means that applying the two displacements to any point gives the same result.