What is the difference between $z=0$ plane and $26z=0$ plane

Question

I'm using this site to calculate a plane equation. The points are $(2,3,0)$, $(5,1,0)$ and $(6,9,0)$. The result is $26z = 0$ plane.

Is there a difference between $26z =0$ and $z = 0$?

Moreover, when I use $(10,2,8)$, $(15,5,0)$ and $(5,11,4)$, the result is $60x + 60y + 60z -1200 = 0$.
But the plane is simply $x+y+z-20 = 0$ plane.

I can see that dividing both sides of the equation gives me the second planes. However, their normals are different and I wonder if some difference occurs when rotating or hifting them?

Answer 1

Always remember:

Computers are very, very stupid.

The plane $26z = 0$ is the exact same one as $z=0$. As it appears, the program does not simplify the expression as far as possible (which a human would be expected to do).

Whenever a computer does something for you, always be prepared to interpret the result, if it seems strange ask why it's so, and always consider whether or not the answer is reasonable.

(That being said, computers are of course wonderful tools, they just need to be used in a correct way.)

Answer 2

There is no difference, as $0/26=0/1$ The solver they use just created a form which wasnt simplified to lowest terms.