Confused about dimension I'm confused cause my intuitive feeling is that if I have a vector (1,2,3) then it's 3D since it has a value for $x$, $y$, $z$, but it's only 1 vector so it can only span a line.
What is the dimension? Cause it's only a line so I feel like it shouldn't have a dimension of 3 but at the same time I wouldn't be able to write that line on a paper.
What's the right way to think about this?
 A: The dimension is a concept we use for spaces or subspaces not for vectors.
More precisely, for a given subspace $S\subset \mathbb R^n $ the dimension of $S$ is, by definition, equal to the number of vectors for a basis of $S$, i.e. the minimum number of independent vectors which span $S$.
A: A vector, strictly speaking, does not have a dimension. The vector $v \equiv (1, 2,3)$ lives in 3D space since it has three components: the x-component, the y-component, the z-component.
We can speak of the line along the vector: $L \equiv \{ (\lambda \cdot 1, \lambda \cdot 2, \lambda \cdot 3) : \lambda \in \mathbb R \}$. A line is 1 dimensional because there's only one direction: the direction along $(1, 2, 3)$.
On the other hand, consider the $xy$ plane: $xy \equiv \{ (x, y, 0) : x, y \in \mathbb R \}$. Here, there are two directions, pointed by the x-axis: $(1, 0, 0)$ and the y-axis: $(0, 1, 0)$. So this is 2 dimensional.
Finally, the full vector space $\mathbb R^3$ is 3-dimensional because it contains three directions: $(1, 0, 0), (0, 1, 0), (0, 0, 1)$.
A: It turns out that the number of vectors in a basis is an invariant.  This number is called the dimension of the vector space.
