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...

perhaps all you really need to say is you have a zero-magnitude vector that is located at a specific point.  technically, it's still a vector, just that its magnitude equals zero.