To calculate a point on a 3D plane, assume that there are 3 non-collinear points Pa, Pb and Pc.

Let Pa=[xa,ya,za], Pb=[xb,yb,zb], Pc=[xc,yc,zc].

The general equation of the plane is of the form

Ax+By+Cz+D=0 ………………… (1)

Consider the figure shown below to calculate the coefficients of the above equation A, B and C

**Note:**** **X refers to cross product

. refers to dot product

* refers to multiplication

The normal to a plane is of the form

N = (Pac X Pab)/(|Pac X Pab|)

N = (Pc-Pa) X (Pb-Pa)/(| (Pc-Pa) X (Pb-Pa)|)

We find that the resulting direction vectors of the normal from the three points Pa, Pb, Pc are

N = [A,B,C] (The coefficients of the equation of the plane in Equation (1))

Now consider an arbitrary point P=[x,y,z] as shown below

It is found that the plane equation satisfies the following property

(P-Pa).N=0

=>A(x-xa)+B(y-ya)+C(z-za)=0

=>Ax+By+Cz+D=0 (where D=-[Axa+Bya+Cza])

=>x=-(1/A).[By+Cz+D]

=>y=-(1/B).[Cz+Ax+D]

=>z=-(1/C).[Ax+By+D]

Thus depending on whether x, y or z is unknown the equation of the plane can be found.

To understand this consider the example where Pa=[1,2,1], Pb=[1,0,2], Pc=[2,2,1] and P=[4,5,z] where z has to found as shown below

To find z follow the below mentioned steps

**Step1:** Find the coefficient terms A, B, C, D

Pac=Pc-Pa=(2-1,2-2,1-1)=(1,0,0)

Pab=Pb-Pa=(1-1,0-2,2-1)=(0,-2,1)

The normal to a plane is of the form

N = (Pac X Pab)/(|Pac X Pab|)

N = (Pc-Pa) X (Pb-Pa)/(|(Pc-Pa) X (Pb-Pa)|)

Pac = (Pc-Pa) = [2,2,1] - [1,2,1] = [1,0,0]

Pab = (Pb-Pa) = [1,0,2] - [1,2,1] = [0,-2,1]

(Pac X Pab)= i j k

1 0 0

0 –2 1

= i(0) - j(1) + k(-2)

(|(Pc-Pa) X (Pb-Pa)|)= ?[(0)2 + (1)2+ (2)2]

=√[1+4]

=√5

Substituting in N = (Pc-Pa) X (Pb-Pa)/(|(Pc-Pa) X (Pb-Pa)|)

N=[0 –1 –2]/√5

Therefore A=0 , B=-1/√5 , C=–2/√5

**Step2:** Find D

D=-[Axa+Bya+Cza]

D=-[0*1 + (-1/√5)*2 + (–2/√5)*1]

=-[0-2/√5-2/√5]

=4/√5

**Step3:** Find z by substituting the value of x=4 and y=5 in point P

z=-(1/C).[Ax+By+D]

z=-(1/–2/√5).[0*x+(-1/√5*y)+4/√5]

z=(√5/2)*[0+(-1/√5)*5+4/√5]

z=(√5/2)*[0+(-5/√5)+ 4/√5]

z=(√5/2)*[-1/√5]

z=-1/2