Chapter 14.4, Problem 82E

To determine

Let $\overrightarrow{r}\left(t\right)=\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)$ be a path with curvature $k\left(t\right)$ and define the scaled path $r_1\left(t\right)=\left(\lambda x\left(t\right)\right),\lambda y\left(t\right),\lambda z\left(t\right)$, where $\lambda \ne 0$ is a extant. Prove that curvatures $k_1\left(t\right)$ of $r_1\left(t\right)\text{ is }{k}_1\left(t\right)={\lambda }^{-1}k\left(t\right)$. This explains why the curvature of a circle of Radius R is proportional to $\frac{1}{R}\left(\text{in fact it is equal to }\frac{1}{R}\right)$.