How to convert a scalar equation to parametric and vector equation

For example, using: x+y+z-10=0

How to write a vector equation for this plane?

Find any three points in the plane (the answer isn't unique). Form two vectors from these and use any one of them to write in the form r = p + su + tv where p is a point on the plane (it's position vector), u and v are vectors in the plane, and s and t are scalar parameters.

Three obvious points are

(10, 0, 0), (0, 10, 0), and (0, 0, 10).

The vectors with initial point (10, 0, 0) and having the other as terminal points are

u = <-10, 10, 0> and v = <-10, 0, 10>

Letting p = <10, 0, 0>, the plane has vector equation

r = <10, 0, 0> + s<-10, 10, 0> + t<-10, 0, 10>, where -∞ < s, t < ∞.