Are all vectors straight lines? Is there a math field that deals with quadratic, cubic etc. vectors? Or a non-linear equivalent of a vector? If so, why are they so much less common than linear vectors?

From your response to my comment in your OP, I'll talk about why vectors are always considered "rays" geometrically.
The most important interpretation of a vector is as a list of numbers (we'll say from $\Bbb R^n$.) The list of numbers determines a point in $\Bbb R^n$, and if you imagine connecting this to the origin with a straight line and adding an arrowhead at the end with the point, you have a "ray" representing the vector. Actually you can slide this ray (without changing the direction) all over space and still have the same vector. Strictly speaking it is not a segment because it has a fixed direction. If you put the arrowhead on the other end, it is a different vector.
Since the vector only carries the information about a single point, it would does not determine anything more complex than a line. In order to "curve" the vector, you'd need to say more about the path it follows on its way from start to finish. 
Another reason is that the points lying on the line determined by a vector through the origin are a subspace of the big space. Subspaces "aren't curved" in some sense. This property causes $\alpha v,\beta v, (\alpha+\beta)v$ to all lie on the same straight line.
Multiplying a vector (ray) in a real vector space by $\lambda \in \Bbb R$ just stretches or contracts the vector. If the vector were curved somehow, then scaling it would change the curviture of the shape. It does not seem desirable that a geometric property such as curviture is ruined merely by scaling the vector, so it doesn't seem very likely to be useful.
