Mathematics

A vector equation describes a curve by giving you $x$ and $y$ coordinates as functions of a parameter $t$—think of it as a particle's position over time. The derivative $\mathbf{r}'(t)$ tells you the instantaneous direction and speed of motion along that curve, so at any point you can draw it as an arrow showing how the curve is "moving." Your job is to visualize both the path itself and how it's being traced at one specific moment.

A vector equation describes a curve by giving you $x$ and $y$ coordinates as functions of a parameter $t$—think of it as a particle's position over time. The derivative $\mathbf{r}'(t)$ tells you the instantaneous direction and speed of motion along that curve, so at any point you can draw it as an arrow showing how the curve is "moving." Your job is to visualize both the path itself and how it's being traced at one specific moment.