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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?

Dan Burrows
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  • What is the definition of a vector you have learned? – Henrik supports the community Jan 21 '15 at 17:37
  • Vector is a much broader term. You could say that Cartesian coordinates are vectors in the real vector space $\Bbb{R}^2$, but there are lots of other vector spaces. – KSmarts Jan 21 '15 at 17:42
  • Cartesian coordinate defines the position of a point w.r.t. the origin of the coordinate system. A vector has both a magnitude and a direction (most likely definition you have learned). – zed111 Jan 21 '15 at 17:43
  • Yeah the definition that I've been going by is the high-school one of direction and magnitude. Thanks for your comments - it's actually a question that is resolved in very few textbooks (that I've seen anyway). – Dan Burrows Jan 21 '15 at 19:08

1 Answers1

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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.

MvG
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  • Thanks MvG, that's a fantastic answer. I think I see what you mean; n-tuples are not the only form of vectors - as you say, certain classes of functions fulfil the axioms of a vector space. Once I find out the difference between a linear space and a vector space, that'll be two of my long-standing ambiguities resolved! – Dan Burrows Jan 21 '15 at 19:05
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    @DanBurrows: What difference? I'd consider linear space and vector space to be synonyms. – MvG Jan 21 '15 at 19:27
  • You're right - I'm thinking of linear operators and linear transformations, for which there is a small distinction. – Dan Burrows Jan 21 '15 at 21:02