Vectors-Linear Systems

[granite wave]

1. Vector equation.
   A plane passing through a point r0 with a normal vector n has the equation:
   n·(r - r0) = 0
   In our case we can take, say, r0 = {0, -4, 0} and n = {3, -1, 2}. So:
   {3, -1, 2}·{x, y + 4, z} = 0
2. Parametric equation.
   We take the equation of a plane as an algebraic equation and solve it like a system of equations.
   3x - y + 2z - 4 = 0
   As the coefficient of y is -1, it's good to take is as the dependent variable.
   y = 3x + 2z - 4
   x and z are independent. So, we get the parametric equations:
   x = x
   y = 3x + 2z - 4
   z = z

Sorry, there was a small mistake. I corrected it.

Oh. Then it's just the parametric equation, just written differently.
In that case:
{x, y, z} = {0, -4, 0} + {1, 3, 0}u + {0, 2, 1}v

Oh, they picked another point. And chose a more difficult vector for some reason. Oh well...

Note that there is an infinite amount of correct answers for a question like this.
You have a choice of infinitely many points r0, for example.

And an infinite amount of choices for the vectors that the plane sits on.

Well, yes, quite a lot of formulas in analytic geometry.