Answer to Question #212387 in Analytic Geometry for Tes

Question

Determine in each case whether the given planes are parallel or perpendicular

a) x + y + 3z 10 = 0 and x + 2y - z = 1

b) 3x - 2y + z - 6 = 0 and 4x + 2y - 4z = 0

c) 3x + y + z - 1 = 0 and -x + 2y + z + 3 = 0

Answer 1

The direction ratio of plane P_i: a_ix+b_iy+c_iy+ d_i=0 is <a_i,b_i,c_i>.

Two planes P₁ and P₂ are parallel if

a₁/a₂ = b₁/b₂ = c₁/c₂ and perpendicular if a₁a₂ +b₁ b₂ + c₁c₂ =0

a). Here <a_1,b₁,c₁>=<1,1,3>and <a_2,b₂,c₃>=<1,2,-1>

a₁/a₂ = 1, b₁/b₂ =1/2, c₁/c₂=-3

a₁a₂ +b₁ b₂ + c₁c₂ =1+2-3=0

Hence these two planes are perpendicular.

b).Here <a_1,b₁,c₁>=<3,-2,1>and <a_2,b₂,c₃>=<4,2,-4>

a₁/a₂ = 3/4, b₁/b₂ =-1, c₁/c₂=-1/4

a₁a₂ +b₁ b₂ + c₁c₂ =12-4-4=4

Hence these two planes are neither parallell nor perpendicular.

c).Here <a_1,b₁,c₁>=<3,1,1>and <a_2,b₂,c₃>=<-1,2,1>

a₁/a₂ = -3, b₁/b₂ =1/2, c₁/c₂=1

a₁a₂ +b₁ b₂ + c₁c₂ =-3+2+1=0

Hence these two planes are perpendicular.