Determine whether the given planes are parallel:

Question

(a) x + 4y - 3z = 2 and 2x - 8y - 6z = 1 ,

(b) 4x - y + 2z = 5 and 12x - 3y + 6z = 8

Answer 1

(a)

x + 4y - 3z = 2

The vector representation of the plane is n1 = <1,4,-3>

2x - 8y - 6z = 1

The vector representation of the plane is n2 = <2,-8,-6>

Condition for parallel planes is n1 = k n2 .  

n2 = <2,-8,-6>

n2 = 2<1,-4,-3>

So n1 ≠ k n2

The vectors are not proportional to each other .

Hence they are not parallel .

Answer 2

(b)

4x - y + 2z = 5 

The vector representation of the plane is n1 = <4,-1,2>

12x - 3y + 6z = 8

The vector representation of the plane is n2 = <12,-3,6>

Condition for parallel planes is n1 = k n2 .  

n2 = <12,-3,6>

n2 = 3<4,-1,2>

so n1= 3 n2

The vectors are proportional to each other .

Hence they are parallel planes .