Question: This is a specific equation of course. It is an example. I am not sure if this was a mistake in the lecture series or some exotic way that I have not the smarts to figure out. Usually it is something like 

$$r = (x,y,z) + t(x\_0,y\_0,z\_0)$$

The first vector being a position vector and the second vector being the direction vector. Then the equation can be parameterized. No problem with that. But this one has me stumped, why would the professor do this? Can anyone guess the motivation or is it a mistake and I am too stupid to figure it out. It is a lecture on Multivariable calculus on u tube. ( I am trying to learn calculus on my own.) It is lecture # 6 at 27:23. ( Just in case someone is wondering why I am asking such a question:-) )

Answer: 
For sure it can.

For example, take two points in space $P\_1$ and $P\_2$, if we construct a line that joins them and choose one parameter $t$, for convenience that ranges $[0,1]$. And impose that for $t=0$ we must obtain point $P\_1$ and for $t=1$ we must obtain the other, we can write without any loss of generality that any point $x$ in between can be expressed:
$$x=t\,P\_2 + (1-t)\, P\_1 \qquad t\in[0,1]\tag{1}$$

This definition can be extended for any value of $t$, defining now a straight line in space, whose explicit form can be obtained from $(1)$
$$x=t\,P\_2 + (1-t)\, P\_1 =P\_1 + (P\_2-P\_1)t \qquad t\in\mathbb{R}$$

This latter equation is the one you are familiar with.
There are many different ways to represent 3D lines. 

What you have here is a parametric form of the line representing all the points given a parameter $t \in \mathbb{R}$ 

* The linear interpolation between two points 
$$ {\bf r} = (1-t)\, {\bf r}\_1 + t\, {\bf r}\_2 $$
* Or you can look at a point and direction
$$ {\bf r} = {\bf r}\_0 + t\, {\bf e} $$

But there are other representations. Without a parameter, but with equation(s) that need to be satisfied

* Point and direction

$$ \frac{x-x\_0}{e\_x} = \frac{y-y\_0}{e\_y} = \frac{z-z\_0}{e\_z} $$

* Direction and moment vectors

$$ \begin{matrix} L : \begin{Bmatrix} {\bf e} \\ {\bf r\_0} \times {\bf e} \end{Bmatrix} & \mbox{or} & L :\begin{Bmatrix} {\bf r}\_2 - {\bf r}\_1 \\ {\bf r}\_2 \times {\bf r}\_1 \end{Bmatrix} \end{matrix} $$

* Intersection of two planes

$$ \begin{matrix} {\bf r} \cdot {\bf n}\_1 = d\_1 \\{\bf r} \cdot {\bf n}\_2 = d\_2 \end{matrix} $$