Show that the points A150 B944 C572 and D311 lie in the same

Show that the points A(1,5,0), B(9,4,-4), C(5,7,-2), and D(3,1,-1) lie in the same plane.

First we need to find the equation of the plane that A, B and C are located in:

n = AB x AC = (8,-1,-4) x (4,2,-2) = (10 , 0 , 20)

So the plane equation is:

10(x-1) + 0(y-5) + 20(z - 0) = 0

x-1 + 2z = 0

So the plane equation is:

x+2z = 1

For point D we have:

x+2z = 3 + 2(-1) = 3-2 = 1

So D lies on the same plane.