Showing if these curves are loops

Question

My Question is :

$\vec{a}(t)= −e^t$i$ + e^t$j

when $−∞ < t ≤ 0$

$\vec{b}(t)= \sqrt{3}t^2$i$+(t^3 − t)$j

when $− 1 ≤ t ≤ 1$

Which of these curves are loops?

How would I go around to show this? Any help will be appreciated

Answer 1

Hint:

We have a loop if there are two different values of the parameter that gives the same point.

For $\vec b(t)$ this means that we have $s \ne t$ such that: $$\vec b (t)=\vec b(s)$$ $$ (\sqrt{3}t^2,t^3-t)=(\sqrt{3}s^2,s^3-s) $$ this means: $$ \sqrt{3}t^2=\sqrt{3}s^2 \iff s=\pm t $$ and, substituting fo the other coordinate the value $s=-t$: $$ s=-t\quad \Rightarrow \quad t^3-t=t-t^3 \quad \Rightarrow \quad 2t(t^2-1)=0 $$

so the values $1$ and $-1$ of the parameter gives the same point and define a loop because for $t \in [-1,1]$ the values of the coordinates are always finite and the curve is continuous.

You can do the same for the other case $\vec a (t)$ and you see that there are no loops here.