Try something like "you can think of a vector as an array of numbers. More precisely, [formal definition here]'. If you give somebody an analogy it gives them s

Try something like "you can think of a vector as an array of numbers. More precisely, [formal definition here]'. If you give somebody an analogy it gives them something to 'hang their thoughts on', if you know what I mean; you read the formalism and try to correlate it back to the analogy. Of course,an analogy is not an equivalence, so there will necessarily be exceptions. You can follow that up with something like "be careful not to assume X from the array of numbers idea; the formal defition states Y, which precludes that". Well, you get the idea.