Answer
Vector equation: $\quad \mathrm{r}(t)=\langle a+(u-a)t, b+(v-b)t, c+(w-c)t \rangle,\quad 0\leq t\leq 1$.
Parametric equations: $\left\{\begin{array}{l}
x= a+(u-a)t\\
y= b+(v-b)t\\
z= c+(w-c)t
\end{array}\right.,\quad 0\leq t\leq 1$.
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 $\quad\mathrm{r}_{0}=\langle a,\ b,\ c\rangle$ and $\displaystyle \mathrm{r}_{1}=\langle\frac{1}{2},\ \displaystyle \frac{1}{3},\ \displaystyle \frac{1}{4}\rangle$,
$\mathrm{r}(t)=(1-t)\mathrm{r}_{0}+t\mathrm{r}_{1},\quad 0\leq t\leq 1$
$=(1-t)\langle a,\ b,\ c\rangle+t\langle u, v, w \rangle,\quad 0\leq t\leq 1$
$=\langle a+(u-a)t, b+(v-b)t, c+(w-c)t \rangle,\quad 0\leq t\leq 1$.
Vector equation: $\quad \mathrm{r}(t)=\langle a+(u-a)t, b+(v-b)t, c+(w-c)t \rangle,\quad 0\leq t\leq 1$.
Parametric equations: $\left\{\begin{array}{l}
x= a+(u-a)t\\
y= b+(v-b)t\\
z= c+(w-c)t
\end{array}\right.,\quad 0\leq t\leq 1$.
