I am working on this problem. Even some of the notation has me confused (the vectors $\vec i$ and $\vec j$).
Let $\vec r(t):=ae^{-bt}\cos(t)\vec i +ae^{-bt}\sin(t)\vec j$ where $a$ and $b$ are positive constants. The trace of $\vec r (t)$ is called the logarithmic spiral.
(a) Show that as $t \to +\infty $, $\vec r (t)$ approaches the origin.
(b) Show that $\vec r (t)$ has finite arc length on $[0,\infty)$.