How to convert a parametric equation to a Cartesian form

Question

I need to convert a plane equation from a parametric form to a Cartesian one. For instance:

(1, 2, -1) + s(1, -2, 3) + t(1, 2, 3)

at

ax+yb+cz+d=0

So basically, my question is: how to find a, b, c and d and what kind of transformation logic.

Answer 1

Calculate the normal vector of this plane:
N = s x t(the vector product of two vectors belonging to the plane)
Now you have the coefficients a, b, c:

N = (a, b, c)

then replace the base point (in the general case, any point on the plane)
(1, 2, -1) to the equation ax + yb + cz + d = 0

a+2b-c+d=0

and find d