Find vector equation for the line defined by $x= 3t + 1, y =5t-2, z=2t + 1$ I was requested to find a vector equation of the line $L$ defined by $x= 3t + 1, y =5t-2, z=2t + 1$. Since I self study --meaning, no teachers to check my answers--, I was wondering if someone could validate my solution. Here is what I attempted.

$\text{Remark}:$ I use comma-separated values inside square brackets "[]" to denote vectors simply because it is easier to write in Latex.
$I$. If $t = 0$, we have that $P_0 = (1, -2, 1)$ is a point of $L$. If $t = 1$, we have that $P_1 = (4, 3, 3)$ is a point of $L$.
$II$. With $\vec{r_1} := [4, 3, 3]$ and $ \vec{r_2} := [1, -2, 1]$, let $$\begin{align} \vec{w} &:= \vec{r_1} - \vec{r_0} \\ &= [3, 5, 2]\end{align}$$
We know $\vec{w}$ is parallel to $L$ insofar it is a representation of the segment $\overline{P_0P_1}$ of $L$.
$III$. A vectorial equation for $L$ is therefore given by $$\vec{g}(t) = \vec{r_1} + t\vec{w} \tag{$t \in \mathbb{R}$} \\ = [4+3t, 3 +5t, 3+2t]$$

I would highly appreciate not only validation/correction ("your proof is right/wrong"), but also any alternative or better ways to try and solve the given problem. Thanks in advance.
 A: Yes, your proof is correct. Here's a direct way to get the line from the question itself. Any point $P$ on the line $L$ can be written as $(x,y,z)$, where $x,y,z$ are given in the question. $[x,y,z]$ is then the position vector pointing to this point $P$ from the origin. Substituting the expressions for $x,y,z$ and splitting,
$$[x,y,z] = [3t+1,5t−2,2t+1] = [3t,5t,2t] + [1,-2,1] = t[3,5,2] + [1,-2,1] = t\vec{w} + \vec{r}$$
The vector equation of the line is just the general position vector for any point on the line; i.e. $t\vec{w} + \vec{r}$ where $\vec{w} = [3,5,2]$ and $\vec{r} = [1,-2,1]$. This is the same as yours but with $t$ shifted by one (substitute $t\gets t+1$ here to verify). The idea used here is similar to the concept of locus of points.
Note: I don't claim that this method will get you full marks in a written exam, but if you become familiar with it, you could quickly create the expression in your head (useful for multiple-choice questions). This method is also useful when your $x,y,z$ are more complicated functions.
