Family of Planes

3 or more Planes are said to form a family if either

  1. The planes are collinear i.e. they pass through a common line of intersection
  2. The planes are parallel to each other
Given any 2 Planes as the following

\(A_1x + B_1y + C_1z + D=0\)
\(A_2x + B_2y + C_2z + D=0\)

The equation of any plane that is either parallel to these 2 planes or is collinear with these 2 planes is given as

\(A_1x + B_1y + C_1z + D + k(A_2x + B_2y + C_2z + D) = 0\)

The value of the variable \(k\) can be found out when some additional input is given. Following are some additional inputs that are generally given
  1. The point through which the resulting plane passes
  2. The plane/axis to which the resulting plane is parallel or perpendicular