from 'Intersection of Three Planes' lesson under vector. i have attached answer.

10 - Intersection of Three Planes MCV4U - Lines and Planes Date:. Intersection of Three Planes How many ways can 3 planes interact? . Pol ( 1 unique soln ) . Line (0o - many soins ) . 2 planes coincide & 35d plane . 3 coinciding planes ( plane ) intersects ( a line ) . Triangular Prism ( no solution ) a parallel planes & 1 intersecting plane (no solution ) 3 parallel planes . 2 coinciding planes & 1 parallel plane (no solution ) ( no solution ) When determining how 3 planes intersect, the first thing to do is look for trivial solutions. The easiest way to find trivial solutions is to look for X co y = o, 2 =0 Page 1 of 410 - Intersection of Three Planes MCV4U - Lines and Planes 10 - Intersection of Three Planes MCV4U - Lines and Planes Example 1: Example 2: How do the following planes interact? How do the planes intersect/interact? x-y+z=-2 2x - y - 2z = -9 3xty- z=-2 x-y+z=1 xty + 2z = 2 x- 5y- z =1 Step 1: Eliminate the same variable twice, leaving us with 2 equations, 2 unknowns. X -4 + 2 = 1 Not the ! 7 10 solution x - y + 2= - 2 ( X( ty + 22 = 2 5) @ ( 2x + 32 = 3) *3 = 6x+ 92 = 9 0 = -2 2x- y - 22 = - 9. @ 4x = -4 7 2 =1 Ean 1: X-y + 20 - 2 2 = -1 5 x + 5y + 102 = 10 - 6x+ 92 = 11 3 x +4- 2 = -2 @ 5x - 32 = -114 - -y + 2 = -2 2 - sy - 2 = 1 > C - sy - 2 = 1 7 Since the original eans 4 = 3 5(-) - 32 = -11 -y = - 3 are not parallel or 2 = 2 -5-32 =-11 Scalar multiples of one - 32 = - 6 y = 3 another , the planes Step 2: 2 = 2 intersect in a triangular prism . Now treat this like a system of equations from grade 10 and solve the system. ONE uni que solution Step 3: Now that you have values for two of the three variables, find the value of the third variable. Example 3: How do the planes intersect/interact? x - y+2z=-1 x - y + 2z = 3 x - 3y +z=0 x-4+22 = -1. Same normal is = ( 1 , - 1 , 2 ) = parallel x- y + 22 = 3 / 2 - 3y + 2=0 no Solution : 2 parallel planes , I intersecting plane Page 2 of 4 Page 3 of 4