Three planes can interact in . Today we will be looking at Type I III and tomorrow we will look at Type IV V. Type 1: Three Parallel Planes When 3 planes are parallel, the 3 normals are Th 1:3 n3 1'52 TI] TI] _\\ n1 "2 A \"3 The equations below represent this situation. We can tell this because all three normal vectors [3,-2,l] , [9,-6,3] > > > > and [6,-4,2] are collinear. That is, 3 m = n; and 2 m = 113. The planes are because their equations do not satisfy these relationships. The planes are distinct, separated because their constant term does not follow the same scalar multiples as the normals. n5 __ "4 7t} > \"4 "l The normal vector of one plane is a of the other 2 normals, but the equations don't follow the same linear combinations. The three normals are still coplanar. n1: 3x2y+zl=0 n1=[3,2,l] ie: \"4; x+3y22+7=O n4=[1,3,2] 35:15x+y22+15=0 n5=[15,l,2] In all three types, each system of equations has because there is no point on all three planes. It is impossible for the coordinates of a point to satisfy all three equations. We say that each system of equations is Describe how the planes in each linear system are related. If there is a unique point or line of intersection, determine its coordinates or equation. Jrlz x+y+z+5=0 a) 712: x+2y+3z+4=0 113: x+y+z=0