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:
[itex]\vec r(t)=<x_{0}, y_{0}, z_{0}> + t<x, y, z>[/itex]
You plug in the various points with what the problem provides. However, a few times I have seen a problem that uses the equation:
[itex]\vec r(t)=<x_{0}, y_{0}, z_{0}> + t<x-x_{0}, y-y_{0}, z-z_{0}>[/itex]

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!