Question

Three planes can intersect in a number of different ways. For each of the combinations below, find the single point of intersection if there is one. If there isn't, explain how the planes do intersect.
a. 3x+4y+2z-1=0 -2x + 5y + 3z + 7 = 0 5x + 4y + 2z − 3 = 0
b. x−2y+3z−1=0 -3x + 5y + 2z + 7 = 0 -x + y + 8z + 5 = 0

Answer 1

a.      planes intersect through a point A(1,2,-5)

1)3x+4y+2z-1=0

2) -2x + 5y + 3z + 7 = 0

3) 5x + 4y + 2z − 3 = 0

(3)-(1)

5x+4y+2z-3-3x-4y-2z+1=0

2x=2

x=1

 

(2)-(1)

-2x+5y+3z+7-3x-4y-2z+1=0

-5x+y+z+8=0

-5+y+z+8=0

y=-3-z

 

(1)

3x+4y+2z-1=0

3+4(-3-z)+2z-1=0

3-12-4z+2z-1=0

2z=-10

z=-5

y=-3+5=2

 

therefore A(1,2,-5)

b. 3 planes do not intersect at a common point,

planes intersect in straight lines: y=11z+4, x=19z+9

1)x−2y+3z−1=0

2) -3x + 5y + 2z + 7 = 0

3) -x + y + 8z + 5 = 0

(3)+(1)

-x+y+8z+5+x-2y+3z-1=0

11z+4=y

(2)-5*(3)

-3x+5y+2z+7+5x-5y-40z-25=0

2x-38z-18=0

x=19z+9