Is the angle between a vector and a line defined? 
Is the angle between a vector and a line defined?

The angle between two lines $a,b$ is defined as the smallest of the angles created.
The angle between two vectors $\vec{a},\vec{b}$ is the smallest angle one of them has to be rotated by so that the directions of $\vec{a},\vec{b}$ are the same.
I have not yet found a definition of the angle between a vector and a line, which makes me wonder if, and if so, how, it is defined.

The question what is the "right" definition here really depends on what the definition of angle between two vectors is. I am not sure which one is meant in the question but it does not matter for the answer in some sense.
There are (at least) two definitions for the angle between two vectors (in the plane), the non-oriented (or unsigned) angle and the oriented (or signed) angle. Both have some merit and relevance.
Depending on which one of the two one choses  for vectors one should in my opinion choose ones definition of angle between a line and a vector, as follows: 

The angle between a vector $\vec{v}$ and a line $L$  is the smaller of the angles between $\vec{v}$ and $\vec{u}$, and $\vec{v}$ and $-\vec{u}$ where $\vec{u}$ is a vetor directing $L$. 

Here the angles is to be taken in the same sense as it is taken for two vectors. Or rather one might define a non-oriented angle and an oriented angle between a vector and a line if one cares about the distinction. 
The definition of "angle" in the question is not compltely precise, so it is hard to tell. But it seems to me that rather the non-oriented angle is meant.

This is not much different from rschwieb's answer, which I upvoted, but the comment thread suggested there is some need for clarification. 
