How to convert Parametric equation to Cartesian form

Question

I need to convert a plane's equation from Parametric form to Cartesian form. For example:

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

to:

ax+yb+cz+d=0

So basically, my question is: how do I find the a, b, c and d, and what's the logic behind the conversion.

Solution

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

N = (a, b, c)

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

a+2b-c+d=0

and find d

Comments

[Thomas Wagenaar]

What is the intuition behind the coefficients of the Cartesian form being equal to the numbers of the normal vector?

[Caridorc]

@ThomasW the only useful vector in determining the orientation of a plane is the vector ortogonal to it (if you use vectors parallel to it, you need 2)