Answer
Vector equation: $\quad \mathrm{r}(t)=\langle-1-2t,\ 2+3t, -2+3t\rangle,\quad 0\leq t\leq 1$.
Parametric equations: $\left\{\begin{array}{l}
x=-1-2t\\
y=2+3t\\
z=-2+3t
\end{array}\right.,\quad 0\leq t\leq 1$.
Work Step by Step
$\mathrm{r}(t)=(1-t)\mathrm{r}_{0}+t\mathrm{r}_{1},\quad 0\leq t\leq 1\quad$ (eq.4 in 12-5)
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Given $\quad\mathrm{r}_{0}=\langle-1,2, -2\rangle$ and $\mathrm{r}_{1}=\langle-3,5,1\rangle$,
$\mathrm{r}(t)=(1-t)\mathrm{r}_{0}+t\mathrm{r}_{1},\quad 0\leq t\leq 1$
$=(1-t)\langle-1,2, -2\rangle+t\langle-3,5,1\rangle,\quad 0\leq t\leq 1$
$=\langle-1+t-3t,\ 2-2t+5t, -2+2t+t\rangle,\quad 0\leq t\leq 1$.
$=\langle-1-2t,\ 2+3t, -2+3t\rangle,\quad 0\leq t\leq 1$.
Vector equation: $\quad \mathrm{r}(t)=\langle-1-2t,\ 2+3t, -2+3t\rangle,\quad 0\leq t\leq 1$.
Parametric equations: $\left\{\begin{array}{l}
x=-1-2t\\
y=2+3t\\
z=-2+3t
\end{array}\right.,\quad 0\leq t\leq 1$.
