What is the difference, if any, between a Cartesian coordinate and a vector? Is it that a vector always has direction and magnitude, whilst Cartesian coordinates do not?
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$\begingroup$Cartesian coordinates are one way to write down vectors as a bunch of numbers. The mathematical concept of a vector space is much broader, so there are many things which are vectors (i.e. which satisfy all the axioms a vector space requires, hence behave like a vector space, hence are a vector space) even though you wouldn't write them down using Cartesian coordinates. One thing that comes to my mind are the functions $\mathbb R\to\mathbb R$ which form an infinite-dimensional $\mathbb R$-vectorspace. Cartesian coordinates are a way to write down a vector by expressing every vector as a linear combination of basis vectors. The existence of a basis is guaranteed for finitely-dimensional vector spaces, but often the choice of basis is pretty arbitrary. Thinking about vectors not too much in terms of coordinates can help reduce reliance on such arbitrary choices.
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