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*"A vector in three-dimensional space. A representation of a vector a=(a1,a2,a3)a=(a1,a2,a3) in the three-dimensional Cartesian coordinate system."*

My question is does representation of a vector to arbitrary precision require 3 values? Can I represent any vector to arbitrary precision with a single value by superimposing a spiral from pole to pole on the surface of a sphere, defining north pole as 0 and south pole as 1, and referencing a point on the spiral as a percentage value. To achieve arbitrary precision I increase the number of "orbits" the superimposed spiral completes around the sphere on its path from north to south. The vector is a ray originating at the center of the sphere, crossing the surface at the referenced point on the spiral. Has this been done before and is it valid?