The vector from point to is . So two vectors in the plane are the vector from (0, 1, 1) to (0, 1, 0), <0- 0, 1- 1, 0- 1>= <0, 0, -1> and the vector from (0, 1, 1) to (-2, -1, -1), <0-(-2), 1-(-1), 1-(-1)>= <2, 2, 2>.
A vector perpendicular to those two vectors, and so perpendicular to the plane is their cross product.
A two dimensional surface in a three dimensional space can be written as a single equation, z= f(x,y) or, more generally, g(x,y,z)= constant. A plane can be written as a linear equation, say, ax+ by+ cz= d for some constants, a, b, c, and d. Parametric equations, in three dimensions, are three equations x= f(t), y= g(t), z= h(t) for a one dimensional figure (a curve or line) or x= f(s, t), y= g(s,t), z= h(s,t) for a two dimensional figure (the number of parameters is the dimension). A plane can be written in linear equations: x= as+ bt+ c, y= ds+ et+ f, z= gs+ ht+ i.
Given a surface given by the single equation, z= f(x,y), you can always take x and y themselves as parameters: x= s, y= t, z= f(s, t).