Plane - Plane Passing Through the Intersection of Two Given Planes

notes

Let π_1 and π_2 be two planes with equations vec r  . hat n _1 = d_1 and vec r . hat n _2 = d_2 respectively.  The position vector of any point on the line of intersection must satisfy both the equations fig.

If vec t  is the position vector of a point on the line , then
vec t . hat n_1 = d_1 and vec t . hat n _2 = d_2
Therefore , for all real values of  λ, we have
vec t . (hat n _1 + lambda hat n_2) = d_1 + lambda d_2
Since vec t is arbitrary, it satisfies for any point on the line.
Hence , the equation vec r . (vec n_1 + lambda vec n_2) = d_1 + lambda d_2   represents a plane π_3 which is such  that if any vector  vec r satisfies both the equations π_1 and π_2, it also satisfies the equation π_3 i.e., any plane passing through the intersection of the planes
vec r . vec n_1 = d_1 and vec r . vec n_2 = d_2
has the equation  vec r . (vec n_1 + lambda vec n_2) = d_1 + lambda d_2    ...(1)

Cartesian form:
In Cartesian system, let  vec n_1 = A_1 hat i + B_2 hat j + C_1 hat k
vec n_2 = A_2 hat i + B_2 hat j + C _2 hat k
and vec r = x hat i + y hat j + z hat k
Then (1) becomes
x (A_1 + lambda A_2) + y(B_1 + lambda B_2) + z(C_1 +lambda C_2) = d_1 + lambda d_2
or (A_1x +B_1y + C_1z -d_1) + lambda (A_2x + B_2y + C_2z -d_2) = 0     ..(2)
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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