Introduction to Pair of Planes

1. Any given General Quadratic Equation in 3 Variables mathematically represents a Pair of Planes if the Rank of it's E-Matrix (denoted by \(R_4\) or \(R_E\) ) is Lesser or Equal to 2.
   
   Depending on the Rank of it's e-Matrix (denoted by \(R_3\) or \(R_e\) ) and whether the Eigen Values of it's e-Matrix are of Same Sign or Not, the Pair of Planes can be of following 5 Types
   1. 2 Co-Incident Real Planes when \(R_E=1\) and \(R_e=1\)
   2. 2 Parallel Real Planes when \(R_E=2\) and \(R_e=1\)
   3. 2 Parallel Imaginary Planes when \(R_E=2\) and \(R_e=1\)
   4. 2 Intersecting Real Planes when \(R_E=2\) and \(R_e=2\) and Eigen Values it's e-Matrix are Not of Same Sign.
   5. 2 Intersecting Imaginary Planes when when \(R_E=2\) and \(R_e=2\) and Eigen Values it's e-Matrix are of Same Sign.
   Whether a General Quadratic Equation in 3 Variables represents 2 Parallel Real Planes or 2 Parallel Imaginary Planes can only be found out after Factoring the Quadratic Equation.
2. A General Quadratic Equation in 3 Variables representing a Pair of Planes can be Factored into it's 2 Constituent Real or Complex Linear Equations as follows
   
   \(A_1x + B_1y + C_1z + D_1=0\)
   
   \(A_2x + B_2y + C_2z + D_2=0\)
   
   The Coefficients \(A_1, A_2, B_1, B_2, C_1\) and \(C_2\) and the Constants \(D_1\) and \(D_2\) can be Real or Complex based on the Type of Pair of Planes as follows
   1. For 2 Co-Incident Real Planes, 2 Parallel Real Planes and 2 Intersecting Real Planes all the Coefficients and Constants are Real.
   2. For 2 Intersecting Imaginary Planes \(A_1\) and \(A_2\), \(B_1\) and \(B_2\), \(C_1\) and \(C_2\), \(D_1\) and \(D_2\) are Complex Conjugates of Each Other.
   3. For 2 Parallel Imaginary Planes \(A_1, A_2, B_1, B_2, C_1\) and \(C_2\) are Real but \(D_1\) and \(D_2\) are Complex Conjugates of Each Other.