Family of Plane - Practice Questions & MCQ

Equation of plane parallel to a given plane

Parallel planes have the same normal vectors, so the equation of plane parallel to is of the form , where d₂ can be determined by using the given conditions.

In cartesian form, if ax + by +cz + d = 0 is given plane, then  the plane parallel to this plane is ax + by + cz + k = 0, where k is any scalar.

Plane passing through the intersection of two given planes

Vector Form

Equation of plane passing through the intersection of planes and is 

Proof:

Let π₁ and π₂ be two planes with equations and respectively. The position vector of any point on the line of intersection must satisfy both the equations.

If  is the position vector of a point P on the line, then 

Therefore, for all real values of λ, we have

Since, is arbitrary, it satisfies for any point on the line.

Hence, the equation represents a plane π₃ which is such that if any vector  satisfies both the equations π₁ and π₂ , it also satisfies the equation π₃ i.e. any plane passing through the intersection of the planes.
 

Cartesian Form

In Cartesian system, let

which is the required Cartesian form of the equation of the plane passing through the intersection of the given planes for each value of λ.