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The Relationship Between Tangents and Vectors
The Greeks and the Chinese used non-curved math to approximate what happens as you get to curvature. The idea of a tangent vector, when we have the rim of my glass, and I talk about the line tangent to the glass, is an analog for differential operators in Newton's calculus. Non-zero vectors which have zero length because we took a word that you know very well and then borrowed it someplace where you really don't have any intuition at all.