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Coulomb's law for magnets only works for finite sized magnets. Thus you cannot get indefinitely close. You can ask the same question for electric charges, such as electrons and protons. As one gets too close, the laws of quantum mechanics must be used in order to properly predict the behavior. For example, see Bohr's atomic model for an early attempt to explain this behavior for the hydrogen atom
Consider electron diffraction experiments, easily repeated with a transmission electron microscope (TEM) and a thin target. For example, with a 10 nm polycrystaline film one sees rings, corresponding to the FCC gold pattern rotated about the beam axis. If the beam intensity is dialed down (lower flux) the same pattern appears, only speckled, as the diffracted electrons strike the detector. If enough time passes, the pattern fills in, and you receive the full diffraction pattern. In this case there a tremendous number of slits from the crystals. Yet the pattern corresponds.
This isn't the case for light: the index of refraction varies by wavelength; and is n=c/v for that media. This accounts for the dispersion of light. The effect is negligible in air and non-existant in vacuum.
The mathematical definition of a vector is more comprehensive than the introductory model provided by an introductory physics class; you will find that there is more to learn even after you take a full linear algebra class! In any case, a vector is an element of a vector space. When viewed against the definitions, the real numbers form a vector space, as does the space of polynomials, etc. etc. The nomenclature of "scalar, vector, tensor" derives from a scheme where tensors are used, and one is seeking the invariants.
The fundamental theorem of linear differential equations tells you that a second order ODE will have two basis functions; for SHO these are Sin and Cos. Linear combinations of these make up the general solution to the homogeneous case. The complex exponential ansatz leads you to the general solution.
@hyportnex: an elementary question requires an elementary answer. Otherwise one is assuming that everybody is a graduate student, and that in order to provide an answer, one must be willing to write an academic thesis!
