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Family of Plane is considered one of the most asked concept.
26 Questions around this concept.
The equation of the plane containing the line
and parallel to the plane,
Let the equation of the plane passing through the line of intersection of the planes be
. For
, if the distance of this plane from the point
is
, then
is equal
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.
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