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In many texts you see the definition of velocity as the time derivative of position. In the context of affine space, position is not a vector quantity. So in the context of affine space, is the correct definition of velocity the time derivative of the displacement instead of the time derivative of the position?

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In affine spaces the derivative of a curve $P=P(t)$ is a vector since couples of points uniquely define vectors: $$P(t + h)-P(t)$$ as a consequence of the affine structure. As it is a well-defined vector it can be multiplied with a number and the velocity is defined as the limit of the vector valued function $$\lim_{h\to 0} h^{-1}(P(t + h)-P(t))\:.$$ Evidently, this limit defines a vector as well.

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  • $\begingroup$ Thanks. Just to clarify, is $P(t+h)-P(t)$ correctly referred to as a displacement vector or translation vector? $\endgroup$ Apr 8, 2020 at 15:16

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