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There's also the case that there's a bunch of vectors you _can't_ represent as tuples of numbers. A vector with orientation but a magnitude of 0 is a completely valid vector afaik, and you could do things with it like normalizing it to get a unit vector of the same orientation, and it's not representable as a tuple.

The tuple model is extremely useful, but incomplete.

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Vectors with different orientation but the same magnitude are the same vector (the zero vector.) Observe that if you add a zero magnitude vector and a unit vector, the result is just the same unit vector. It follows that the normalization scheme you described can't be a function on vectors (since you can "normalize" two equal vectors but get two different results.)

> A vector with orientation but a magnitude of 0 is a completely valid vector afaik

Not by the mathematicians' definition. Maybe somewhere in physics such objects are useful, but if you formalized them you'd get something other than a vector space.

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