Find the domain of the vector function $\vec{r}(t)=\left\langle t^{3}, \sqrt{t+3}, \sqrt{-1-t}>\right.$ Domain: $\{t \mid \square \leq t \leq \square\}$

Step 1 ：The vector function \(\vec{r}(t)=\left\langle t^{3}, \sqrt{t+3}, \sqrt{-1-t}\right\rangle\) is defined only when \(t+3\geq 0\) and \(-1-t\geq 0\).

Step 2 ：From \(t+3\geq 0\), we get \(t\geq -3\).

Step 3 ：From \(-1-t\geq 0\), we get \(t\leq -1\).

Step 4 ：Hence, the domain of \(\vec{r}(t)\) is \(\boxed{[-3,-1]}\).