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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.
Note that vectors themselves are free floating - allowing addition, etc.
