URGENT!!! Find the equation of a plane passsing through 4pts.?

Question

4pts.
(x,y,z)
(x1,y1,z1)
(x2,y2,z2)
(x3,y3,z3)
Find the equation of the plane passing through all 4pts. NEED HELP ASAP...

Answer 1

It only takes three points to uniquely define a plane provided they are not collinear.

P1(x1,y1,z1)
P2(x2,y2,z2)
P3(x3,y3,z3)

Create two directional vectors from the points.

u = P1P2 = P2 - P1 =
v = P1P3 = P3 - P1 =

Take the cross product of the two vectors to get the normal vector to the plane.

u X v = X
= [(y2-y1)(z3-z1) - (y3-y1)(z2-z1)]i
+ [(x3-x1)(z2-z1) - (x2-x1)(z3-z1)]j
+ [(x2-x1)(y3-y1) - (x3-x1)(z2-z1)]k

Now plug in a point and write the equation of the plane. Let's choose P1.

[(y2-y1)(z3-z1) - (y3-y1)(z2-z1)]*(x - x1)
+ [(x3-x1)(z2-z1) - (x2-x1)(z3-z1)]*(y - y1)
+ [(x2-x1)(y3-y1) - (x3-x1)(z2-z1)]*(z - z1) = 0
_______

By the way, it's problematic calling a point (x,y,z) because those are the variables used to denote a general point in the equation. However, the plane is uniquely defined by the points P1, P2, and P3 provided they are not collinear.