$v_0$ and $v_p$ are vectors.
Let $v_0, v_1$ and $v$ be vectors, all emanating from $(0, 0, 0)$. Suppose the line $l$ is passing through their heads. Let $v_p$ be on the line $l$ such that $v_1 = v_0 - v_p$(head-to-tail method). Extend $v_p$ with $t$ in such way that the head of $tv_p$ ends where $v$ meets $l$. Then by head-to-tail, $v = v_1 + tv_p$.
I don't know if that makes sense. If it doesn't, how do I fix it?
Thanks.
It seem the following,
The vector equation of the line $l$ through the head of $v_0$ and parallel to $v_p$ is $v_0+tv_p$, because a vector $v$ belongs to the line $l$ iff $v-v_0$ is collinear to $v_p$, that is $v-v_0=tv_p$ for some real number $t$.