is there a format math object ........

A vertex is a point in space.  A vector is a direction, but has no position.

Is there a formal mathematical object that contains both a vertex and a vector?  Lines aren't what I have in mind (those are defined by two vertices).  Right now in my program I'm just calling this construct a "bone" but that's a stupid word for it... is there a more formal terminology for a [Vertex, Vector] tuple?  "Directional point"?  Nah...

also, if a point has a direction, that would imply that it can have an orientation.  now, if the point is the classical zero-dimensional thing, can it be assigned an orientation?  now I'm thinking of how a point on a curve can be assigned a slope value, though that idea has been in terms of the curve having a specific slope at a given point, rather than in terms of a point itself having a slope.