Definitions
One problem with math pedagogy is that it often tries to describe math things by analogy. This is bad because analogies are specific and limited.
A VECTOR IS A LIST OF NUMBERS
You got that?? They don’t have “magnitude and direction”, they don’t represent “length, breadth, and height”, they don’t have arrow at the ends of them that point in a certain direction. All they are are lists of numbers. Each number in that list represents one dimension in the infinity of possible vector spaces.
What you can do with those lists, however, is very very interesting…