Final answer:
Reparametrizing the curve r(t)=(5+3t)i + (-3-3t)j + (-2-2t)k in terms of arc length requires finding the curve's speed, integrating it to obtain the arc length function, solving for t in terms of s, and substituting back into the original equation.
Step-by-step explanation:
The process of reparametrizing the curve r(t)=(5+3t)i + (-3-3t)j + (-2-2t)k in terms of arc length involves finding the integral of the curve's speed to get the arc length function, and then solving for t as a function of s, the arc length. First, calculate the speed of the curve, which is the magnitude of the velocity vector v(t). The speed is given by v(t) = √(3i - 3j - 2k)², which simplifies to v(t) = √(9 + 9 + 4), or v(t) = √22. The arc length s from the starting point to a point at time t is the integral from 0 to t of the speed: s(t) = √22 · t. Solving for t yields t(s) = s / √22. Substituting back into the original curve equation gives the reparametrized curve in terms of arc length.