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I was wondering, when you take the magnitude of the vector function $r(t)$, what does it represent geometrically?

Does it represent the magnitude of the displacement vector, whose initial point is the origin, and terminating point is a point on the curve sketched by the vector function at a particular t-value?

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Yes.

The components of $r$ in the dimensions given the projections of the vector on the respective axis. The 2 norm or the magnitude of this function would represent how far (Euclidean) you are from the origin at time $t$.

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