Do one-dimensional vectors exist? What are they used for? I'm taking a linear algebra course this semester, and so far we've only talked about vectors in $\mathbb{R}^2, \mathbb{R}^3$ and higher dimensions. 
Does it makes sense to talk about one-dimensional vectors in $\mathbb{R}^1$?
Since we visualize $\mathbb{R}^2$ as a plane, would $\mathbb{R}$ be a simple number line?
If one-dimensional vectors are a thing, what are they used for?
 A: Yes. Not only are one dimensional vectors a thing, "zero dimensional" vectors are too! An example of a one dimensional vector would just be any real number, as you observed. A zero dimensional vector would be an element of the trivial vector space $\{0\}$.
This might seemingly conflict with uses of "vector" one typically learns in middle/high school, which states that a vector is a quantity with magnitude and direction, and quantities with only magnitude are called scalars. You can still think of elements of $\mathbb R$ this way, in drawing an arrow from $0$ to any $x$ on the real number line and having "right or left" (positive or negative) be your direction. But more generally, this is not actually a distinction we make in math, and the definition of a vector space over a field $K$ is abstract in terms of the defining axioms. The abstract definitions may mean that we call some weird things vectors that are hard to get used to at first, but eventually you'll realise the usefulness of our definitions.
A: The answer to your first question is yes; you may, if you wish, call any $a \in \mathbb{R}$ a vector. However, there isn’t much purpose, since they fulfill exactly the same purpose as what you may wish to call a scalar. For instance, the vector $<3>$ and the scalar $ 3 $both produce the same vector if you take their dot product / product with any vector in $\mathbb{R}^n$.  
To answer your third question, they aren’t really a ‘thing’ for the exact reason stated above that there is no real use for calling them vectors. 
To answer your second question, the space you are considering is indeed the number line.
