given planes

given planes Theorem . The vector equation of a plane which passes through the line of intersection of the planes r n r n is given by ( r n r n , where λ ∈  . Proof Consider the equation r n r n ... ( ) The above equation can be simplified as ) ( ...

( ) Put  , Then the equation ( ) becomes r n ... ( ) The equation ( ) represents a plane. Hence ( ) represents a plane. Let r  be the position vector of any point on the line of intersection of the plane.

Then r  satisfies both the equations r n r n . So, we have r n  = d ... ( ) and r n  = ... ( ) By ( ) and ( ), r  satisfies ( ).

So, any point on the line of intersection lies on the plane ( ). This proves that the plane ( ) passes through the line of intersection. The cartesian equation of a plane which passes through the line of intersection of the planes a x b y c z a x b y c z is given by a x b y c z a x b y c z Example . Find the equation of the plane passing through the intersection of the planes