Show that the planes 2x – y + 6z = 5 and 5x – 2.5y + 15z = 12 are parallel.
Formula : Plane = r . (n) = d
Where r = any random point
n = normal vector of plane
d = distance of plane from origin
If two planes are parallel , then their normal vectors are either same or proportional to each other
Therefore ,
Plane 1 : - 2x – y + 6z = 5
Normal vector (Plane 1) = (2i – j + 6k) …..(1)
Plane 2 : - 5x – 2.5y + 15z = 12
Normal vector (Plane 2) = (5i – 2.5j + 15k) …..(2)
Multiply equation(1) by 5 and equation(2) by 2
Normal vector (Plane 1) = 5(2i – j + 6k)
= 10i – 5j + 30k
Normal vector (Plane 2) = 2(5i – 2.5j + 15k)
= 10i – 5j + 30k
Since, both normal vectors are same .Therefore both planes are parallel