A plane has equation

.

Substituting your points gives

.

This gives the system

(I switched order of the rows)

You can see so .

You can see

.

You can see

.

is a free variable, you can let it be whatever you like to get a consistent solution set. So why not let .

Therefore a correct equation of the plane is

.

The vectors normal to the plane has the same coefficients as the coefficients of the plane itself.

Therefore a vector normal to the plane is

.

To find the unit vector, divide by its length.

.