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 same

Therefore ,

Parallel Plane x – 2y + 2z – 3 = 0

Normal vector = (i - 2j + 2k)

∴ Normal vector of required plane = (i - 2j + 2k)

Equation of required planes r . (i - 2j + 2k) = d

In cartesian form x – 2y + 2y = d

It should be at unit distance from point (1,1,1)

Distance

For + sign = > 3 = 1 - d = > d = - 2

For - sign = > 3 = - 1 + d = > d = 4

Therefore equations of planes are : -

For d = - 2 For d = 4

x – 2y + 2y = d x – 2y + 2y = d

x – 2y + 2y = - 2 x – 2y + 2y = 4

x – 2y + 2y + 2 = 0 x – 2y + 2y – 4 = 0

Required planes = x – 2y + 2y + 2 = 0

x – 2y + 2y – 4 = 0