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.

You could just take a vector pointing along the direction of the line and use the angle between the original vector and this vector. 
Of course, you would need to choose the orientation your new vector should point: this will affect the angle you get. I would expect you would choose the one which minimizes the angle.
