Q:

A vector or a set of vectors

Eigenvalue problem:
Ax = $\lambda$x
Why is x defined as a single vector (eigenvector)? I would think of it rather as a set of three vectors, each in a different dimension. 
$$\begin{bmatrix}1\\1\\1\end{bmatrix}$$ 
Could this "vector" not be interpreted as a point one unit from the origin in three dimensions? such that it's span would create a one dimensional subspace of $ℝ^3$ passing through that point;
Each of these values have both magnitude and direction, so is a set of vectors also a vector? Or is an eigenvector just a set of vectors? 
If I was given something of the above form, and was told that it was a vector space, I would interpret it as a space (defined by those VECTORS in the set) closed under vector addition and scalar multiplication.
What am I missing? 

A:

$$A x=\lambda x$$ is an equation in the unknowns $\lambda$, a scalar, and $x$, a vector. Obviously, if $x$ is a solution, so is $\mu x$, and the complete solution set is a pencil of straight lines.
When you solve the equation, you take one representative Eigenvector per Eigenvalue, that's enough (degeneracy omitted). In certain cases, you will normalize the representative to unit length to make it unique.
The need to explicitly represent the whole set of collinear vectors does not arise in practice.

