I am studying arclength for a path c(t)=(x(t),y(t),z(t)). I understand that we integrate the derivative of speed etc. My question is, geometrically, what is |c(t)|? c(t) is a formula of displacement, so my naive interpretation is that taking its magnitude would yield distance travelled, but this clearly isn't the case. As a follow-up, what is the difference between the rate of change of displacement and the rate of change of distance?

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$$\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$$

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