Family of Planes

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

The planes are collinear i.e. they pass through a common line of intersection
The planes are parallel to each other

Adding equation of 2 Planes always gives an equation of a Plane that belongs to the same family as of original 2 Planes.

Subtracting equation of 1 Plane from other also gives an equation of a Plane that belongs to the same family as of original 2 Planes. (except in case of 2 parallel Planes whose corresponding \(x\), \(y\) and \(z\) coefficients are same).

Given any 2 Planes as the following

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

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

The point through which the resulting plane passes
The plane/axis to which the resulting plane is parallel or perpendicular

Related Topics and Calculators

Intorduction to Planes, Derivation/Representation of Equation of Planes, Finding Points on Plane/Intercepts of Plane, Types of Planes, Condition for Coplanarity of 4 Points, Projection of Vector on a Plane, Angular Normal of a Plane, Angle Between 2 Planes, Angle Between a Line and a Plane, Relation Between a Line and a Plane, Relation Between 2 Planes, Relation Between 3 Planes, Condition for Collinearity and Concurrency of Planes,

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