Varying Forms for an Equation of a Line in R^3

[kieth89]

I was doing some course work earlier today and noticed that I've seen two different equations for a line in 3D space. Usually the equation I use is:
$\vec r(t)=<x_{0}, y_{0}, z_{0}> + t<x, y, z>$
You plug in the various points with what the problem provides. However, a few times I have seen a problem that uses the equation:
$\vec r(t)=<x_{0}, y_{0}, z_{0}> + t<x-x_{0}, y-y_{0}, z-z_{0}>$

How do I know which equation to use? Or are these equivalent?

Thanks!

 

[geoffrey159]

You can define a line either with 1 point and a directing vector, or with 2 points.
The first form of is the definition of a line passing through $M_0 = (x_0,y_0,z_0)$ and directed by $\vec u = (x,y,z)$, while the second form is the definition of a line passing through $M_0$ and $M = (x,y,z)$ (directed by $\vec{ M_0M}$)

 

[kieth89]

Oh, that makes perfect sense. Thank you!