"A vector is a composite quantity that has a magnitude and a direction. If something has a direction component and/or a magnitude component, it can be represented by a vector, and if something is a vector, we can ask for its direction or its magnitude (or both.)"
Cool. That works well, if a bit geometrical. I'm going to think about rewording the intro in this way.
However, I'd still have to introduce the notion of "coordinate vector" somehow, which begs for the word "tuple" or array, neither of which the right choice.
What I've described is the concept of a vector. You will have to determine the notation, and in setting the notation, and ordered list of basis vector coefficients is most commonly called a tuple, which is a generalization of ordered pair / triple / quadruple, etc.. (Though 'tuple' doesn't usually imply ordered.)
The reason 'array' doesn't really work is that arrays are generally understood to be homogeneous in type, while tuples are usually Cartesian products. So if you are using a Cartesian coordinate system, tuple is the ideal term.
