Intersection of 3 Planes
|This example is the same as the previous example
except that P3 has changed.
P1 x - y + 4z = 5 => n1 = (1,-1,4)
P2 3x + y + z = 3 => n2 = (3, 1,1)
P3 5x - y + 9z = 13 => n3 = (5,-1,9)
As before, L12 has m = (-5,11,4) and contains point (2,3,0). This time the point lies on P3
The 3 planes do have a common intersection
|Now consider the following 3 planes.
x + y + z = 1 => n1
Analysis of normal vectors
For a point on L12 let z = 0
Sub L12 into P3
n1 x n2 = m direction vector for the line of intersection of P1 and P2
if m · n3 = 0 either no solution or a line of intersection
if m · n3 <> 0 then a unique intersection point.