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My logic is as follows...

The purpose of mathematics is to describe reality. A velocity vector for example should tell me what I need to know about the direction and speed of an object. If the object is a ball sitting still in the middle of a field a velocity magnitude of zero tells me everything I need to know. If the object is a bullet with a velocity magnitude of zero I am still very much interested in which direction the bullet is pointed.

Based on this reasoning it seems to me that there are 2 classes of vectors, those which do and those which do not still have a direction if their magnitude is zero.

Is this reasoning in line with how vectors are usually used? If so, what kind of terminology and notation is used to distinguish the 2? If not, how should the difference be described mathematically