Answer
(a) The statement is not true.
(b) The statement is true.
(c) The statement is true.
Work Step by Step
(a) We have two paths ${{\bf{r}}_1}\left( t \right)$ and ${{\bf{r}}_2}\left( t \right)$ parametrized by $t$. However, we cannot assume that the parameter values ${t_1}$ and ${t_2}$ are equal at the point of intersection. Thus, if ${{\bf{r}}_1}\left( t \right)$ and ${{\bf{r}}_2}\left( t \right)$ intersect, they do not necessarily collide. The statement in (a) is not true.
(b) If ${{\bf{r}}_1}\left( t \right)$ and ${{\bf{r}}_2}\left( t \right)$ collide, then ${{\bf{r}}_1}\left( {{t_0}} \right) = {{\bf{r}}_2}\left( {{t_0}} \right)$ at some time ${t_0}$. Thus, the two curves intersect. The statement in (b) is true.
(c) Since intersection involves point in the curves, it is an intrinsic property, that is, it depends on the geometric property of the two curves. Since parametrization of curves is not unique, therefore, collision depends on the actual parametrizations. The statement in (c) is true.
