First, I would like to clarify what I mean by "intuitive definition": an intuitive definition is an informal understanding of a concept which helps to build mental agility, with the possible side-effect of crippling one's ability to imagine the full scope of the concept.
Next, I would like to clarify what I mean by "effective intuitive definitions": effective intuitive definitions are sets of intuitive definitions, which, taken together, try to minimize the overall crippling effect of intuitive understandings on imagination.
In this document, in section 1.1 the author writes:
I’d say the best intuitive deﬁnition of a vector is “anything that can be represented by arrows that add head-to-tail.”
I am not convinced that is the best intuitive definition. I think that at the very least, for any concept, the best intuitive definition must be a set of intuitive definitions that is an effective intuitive definition.
In particular I feel that the above proposed "best intuitive definition of a vector" is severely cripples the imagination because it ties down one's imagination to three dimensional space.
What are some effective intuitive definitions of vectors?