# What is the relationship between cartesian vectors and tuples?

As far as I have understood the cartesian product, $$\mathbb{R}^2 = \{(x,y) \ \vert \ x \in \mathbb{R} \ \land \ y \in \mathbb{R} \}$$, contains ordered pairs, represented as tuples.
But think I have also seen $$\vec{v} \in \mathbb{R}^2$$.

Firstly, is that correct?
Secondly, does that mean that the elements of $$\mathbb{R}^2$$ can be represented as vectors as well?
In general, what is the relationship between cartesian vectors and tuples?

Tuples (in general points in $$R^n$$) are points in space. Vectors have a length and direction. A one to one correspondence between vectors and points in space can be made by fixing the tail of a vector to the origin and the head of the vector at that point in space.