Wikipedia:Reference desk/Archives/Mathematics/2022 January 5

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January 5

What is special about Cartesian coordinates?

In the context of Euclidean space, the Cartesian coordinate system seems special, even among orthogonal coordinate systems. For example, one can sum vectors by separately summing each component. My intuition is that this specialness is due to the Cartesian coordinate hypersurfaces being linear subspaces of Eulcidean space. Am I right? If so, is there a name for this property? -Amcbride (talk) 02:17, 5 January 2022 (UTC)

In any vector space with a basis one can add vectors by component-wise addition of their representations as coordinate vectors. The basis does not have to be formed by Cartesian-coordinate unit vectors (an orthonormal basis) or even be orthogonal. For a coordinate system based on any basis of a Euclidean space viewed as a vector space, not necessarily Cartesian, the coordinate hypersurfaces are again Euclidean spaces. The hyperbolic paraboloid is a non-Euclidean doubly ruled surface. One can impose a coordinate system in which the "rules" are the coordinate lines. So the implication <coordinate vector system describes Euclidean space> → <coordinate hypersurfaces are Euclidean> is one-way only.  --Lambiam 07:57, 5 January 2022 (UTC)

Thank you! I might understand. So are Cartesian coordinates the intersection of orthogonal coordinates and basis vector representations? -Amcbride (talk) 15:48, 5 January 2022 (UTC)

Properly speaking, it is not the coordinates that are orthogonal, but the set of vectors that are the basis: they are pairwise orthogonal, which is commonly called an "orthogonal basis". But, moreover, the length of each vector must be $1$, and then we have an "orthonormal basis". The set of $n$ Cartesian coordinate vectors $(1,0,...,0),(0,1,...,0),...,(0,0,...,1)$ for $n$-dimensional Euclidean space forms an orthonormal basis, and the change of basis to any other orthonormal basis for that space is an isometric transformation, so the new coordinate system is just as "Cartesian" as the original.  --Lambiam 22:42, 5 January 2022 (UTC)

I think I understand. Thanks again! -Amcbride (talk) 23:14, 5 January 2022 (UTC)