$t = $ 0.00
$\Big\rbrace$ start & end points
$\definecolor{myred}{RGB}{230,0,0}\color{myred}\mathbf{r}_0 = \langle 3,-2,4 \rangle$,
$\definecolor{myblue}{RGB}{0,0,230}\color{myblue}\mathbf{r}_1 = \langle 2,4,-1 \rangle$
$\boxed{\mathbf{r}(t) = (1-t)\mathbf{r}_0 + t\,\mathbf{r}_1,\quad 0\le t\le 1}$
You can think of $t$ as giving a percentage, namely the percentage $\mathbf{r}(t)$ has moved away from $\mathbf{r}_0$ and towards $\mathbf{r}_1$.
When $t = $ 0.00, $\mathbf{r}(t)$ is 100% towards $\color{myred}\mathbf{r}_0$ and
0% towards $\color{myblue}\mathbf{r}_1$.