The goal is to reparametrize the curve r(t) in terms of arc length from a given starting point by computing the derivative's magnitude, integrating it to get arc length, and then inverting this relationship to express the curve in the arc length parameter.
The question pertains to reparametrizing a given curve r(t) = (−1−2t)i + (2+0t)j + (1+0t)k starting from the point (−1,2,1) in terms of arc length. To reparametrize the curve, we first need to compute the magnitude of the curve's derivative to find the speed, and then integrate the speed to find the arc length as a function of t. Once the arc length s(t) is known, we can find its inverse to get t(s) and substitute back into r(t) to express the curve as r(t(s)). However, without the full context and the rest of the curve equation, we cannot provide an exact reparametrization in this case.