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Factoring a Pair of Planes into Linear Equations

1. Once a General Quadratic Equation in 3 Variables has been identified as representing a Pair of Planes, it can be Factored into it's 2 Constituent Real or Complex Linear Equations.
2. Let consider a General Quadratic Equation in 3 Variables representing a Pair of Planes given as follows
   
   $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + K=0$   ...(1)
   
   which has to be Factored into it's corresponding 2 Linear Equations given as follows
   
   $A_1x + B_1y + C_1z + D_1= 0$   ...(2)
   
   $A_2x + B_2y + C_2z + D_2= 0$   ...(3)
   
   Now, the Coefficients and the Constant of the equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations (2) and (3) above as follows
   
   $A= A_1A_2$   ...(4)
   
   $B= B_1B_2$   ...(5)
   
   $C= C_1C_2$   ...(6)
   
   $D= A_1B_2 + A_2B_1$   ...(7)
   
   $E= A_1C_2 + A_2C_1$   ...(8)
   
   $F= B_1C_2 + B_2C_1$   ...(9)
   
   $G= A_1D_2 + A_2D_1$   ...(10)
   
   $H= B_1D_2 + B_2D_1$   ...(11)
   
   $I= C_1D_2 + C_2D_1$   ...(12)
   
   $K= D_1D_2$   ...(13)
   
   The following gives the steps of Finding the Values of the Coefficients and Constants $A_1, A_2, B_1, B_2, C_1, C_2, D_1$ and $D_2$ of the Linear Equations (2) and (3) above
   1. Normalize the General Quadratic Equation in 3 Variables given in equation (1).
   2. If the Coefficient of $x^2$ ($A$) in the equation (1) is Not Zero, then the Coefficient of $x^2$ ($A$) in the Normalized version of equation (1) becomes $1$. Since $A=A_1A_2$, the Values of the Coefficients $A_1$ and $A_2$ can be set as $1$ as well. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A = 1$   ...(14)
      
      $B= B_1B_2$   ...(15)
      
      $C= C_1C_2$   ...(16)
      
      $D= B_1 + B_2$   ...(17)
      
      $E= C_1 + C_2$   ...(18)
      
      $F= B_1C_2 + B_2C_1$   ...(19)
      
      $G= D_1 +D_2$   ...(20)
      
      $H= B_1D_2 + B_2D_1$   ...(21)
      
      $I= C_1D_2 + C_2D_1$   ...(22)
      
      $K= D_1D_2$   ...(23)
      
      Now, a Quadratic Equation can be formed by using the Values of Coefficient $G$ and Constant $K$ in equations (20) and (23) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Gx + K=0$   ...(24)
      
      $\Rightarrow x^2 - (D_1 + D_2)x + D_1D_2=0$   ...(25)
      
      The Values of Constants $D_1$ and $D_2$ can be found out Finding the Roots of Quadratic Equation (24) above.
      
      Now, if Values of either $D_1$ or $D_2$ or Both are Not Zero then the Values of Coefficients $B_1$ and $B_2$ can be found out by Solving the System of Linear Equations given by equations (7) and (11) above as follows
      
      $A_1B_2 + A_2B_1=D$   ...(from equation 7)
      
      $B_1D_2 + B_2D_1=H$   ...(from equation 11)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      $\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}D \\ H\end{bmatrix}$   ...(26)
      
      $\Rightarrow \begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}D \\ H\end{bmatrix}$   ...(27)
      
      Also, the Values of Coefficients $C_1$ and $C_2$ can be found out by Solving the System of Linear Equations given by equations (8) and (12) above as follows
      
      $A_1C_2 + A_2C_1=E$   ...(from equation 8)
      
      $C_1D_2 + C_2D_1=I$   ...(from equation 12)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      $\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}E \\ I\end{bmatrix}$   ...(28)
      
      $\Rightarrow \begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}A_2 & A_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}E \\ I\end{bmatrix}$   ...(29)
      
      Otherwise, if Values of Both $D_1$ and $D_2$ are Zero, a Quadratic Equation can be formed by using the Values of Coefficients $D$ and $B$ in equations (17) and (15) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Dx + B=0$   ...(30)
      
      $\Rightarrow x^2 - (B_1 + B_2)x + B_1B_2=0$   ...(31)
      
      The Values of Coefficients $B_1$ and $B_2$ can be found out Finding the Roots of Quadratic Equation (30) above.
      
      Similarly, a Quadratic Equation can be formed by using the Values of Coefficients $E$ and $C$ in equations (18) and (16) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Ex + C=0$   ...(32)
      
      $\Rightarrow x^2 - (C_1 + C_2)x + C_1C_2=0$   ...(33)
      
      The Values of Coefficients $C_1$ and $C_2$ can be found out Finding the Roots of Quadratic Equation (32) above.
      
      The values of $B_1, C_1$ and $B_2, C_2$ can be matched and verified by evaluating equation (19) above.
   3. If the Coefficient of $x^2$ ($A$) in the equation (1) is Zero and Coefficient of $y^2$ ($B$) in the equation (1) is Not Zero, then the Coefficient of $y^2$ ($B$) in the Normalized version of equation (1) becomes $1$. Since $B=B_1B_2$ , the Values of the Coefficients $B_1$ and $B_2$ can be set as $1$ as well. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= A_1A_2$   ...(34)
      
      $B= 1$   ...(35)
      
      $C= C_1C_2$   ...(36)
      
      $D= A_1 + A_2$   ...(37)
      
      $E= A_1C_2 + A_2C_1$   ...(38)
      
      $F= C_1 + C_2$   ...(39)
      
      $G= A_1D_2 + A_2D_1$   ...(40)
      
      $H= D_1 + D_2$   ...(41)
      
      $I= C_1D_2 + C_2D_1$   ...(42)
      
      $K= D_1D_2$   ...(43)
      
      Now, a Quadratic Equation can be formed by using the Values of Coefficient $H$ and Constant $K$ in equations (41) and (43) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Hx + K=0$   ...(44)
      
      $\Rightarrow x^2 - (D_1 + D_2)x + D_1D_2=0$   ...(45)
      
      The Values of Constants $D_1$ and $D_2$ can be found out Finding the Roots of Quadratic Equation (44) above.
      
      Now, if Values of either $D_1$ or $D_2$ or Both are Not Zero then the Values of Coefficients $A_1$ and $A_2$ can be found out by Solving the System of Linear Equations given by equations (7) and (10) above as follows
      
      $A_1B_2 + A_2B_1=D$   ...(from equation 7)
      
      $A_1D_2 + A_2D_1=G$   ...(from equation 10)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      $\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}D \\ G\end{bmatrix}$   ...(46)
      
      $\Rightarrow \begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}D \\ G\end{bmatrix}$   ...(47)
      
      Also, the Values of Coefficients $C_1$ and $C_2$ can be found out by Solving the System of Linear Equations given by equations (9) and (12) above as follows
      
      $B_1C_2 + B_2C_1=F$   ...(from equation 9)
      
      $C_1D_2 + C_2D_1=I$   ...(from equation 12)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      $\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}F \\ I\end{bmatrix}$   ...(48)
      
      $\Rightarrow \begin{bmatrix}C_1 \\ C_2\end{bmatrix}=\begin{bmatrix}B_2 & B_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}F \\ I\end{bmatrix}$   ...(49)
      
      Otherwise, if Values of Both $D_1$ and $D_2$ are Zero, a Quadratic Equation can be formed by using the Values of Coefficients $D$ and $A$ in equations (37) and (34) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Dx + A=0$   ...(50)
      
      $\Rightarrow x^2 - (A_1 + A_2)x + A_1A_2=0$   ...(51)
      
      The Values of Coefficients $A_1$ and $A_2$ can be found out Finding the Roots of Quadratic Equation (50) above.
      
      Similarly, a Quadratic Equation can be formed by using the Values of Coefficients $F$ and $C$ in equations (39) and (36) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Fx + C=0$   ...(52)
      
      $\Rightarrow x^2 - (C_1 + C_2)x + C_1C_2=0$   ...(53)
      
      The Values of Coefficients $C_1$ and $C_2$ can be found out Finding the Roots of Quadratic Equation (52) above.
      
      The values of $A_1, C_1$ and $A_2, C_2$ can be matched and verified by evaluating equation (38) above.
   4. If the Coefficients of Both $x^2$ ($A$) and $y^2$ ($B$) in the equation (1) are Zero and Coefficient of $z^2$ ($C$) in the equation (1) is Not Zero, then the Coefficient of $z^2$ ($C$) in the Normalized version of equation (1) becomes $1$. Since $C=C_1C_2$ , the Values of the Coefficients $C_1$ and $C_2$ can be set as $1$ as well. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= A_1A_2$   ...(54)
      
      $B= B_1B_2$   ...(55)
      
      $C= 1$   ...(56)
      
      $D= A_1B_2 + A_2B_1$   ...(57)
      
      $E= A_1 + A_2$   ...(58)
      
      $F= B_1 + B_2$   ...(59)
      
      $G= A_1D_2 + A_2D_1$   ...(60)
      
      $H= B_1D_2 + B_2D_1$   ...(61)
      
      $I= D_1 + D_2$   ...(62)
      
      $K= D_1D_2$   ...(63)
      
      Now, a Quadratic Equation can be formed by using the Values of Coefficient $I$ and Constant $K$ in equations (41) and (43) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Ix + K=0$   ...(64)
      
      $\Rightarrow x^2 - (D_1 + D_2)x + D_1D_2=0$   ...(65)
      
      The Values of Constants $D_1$ and $D_2$ can be found out Finding the Roots of Quadratic Equation (64) above.
      
      Now, if Values of either $D_1$ or $D_2$ or Both are Not Zero then the Values of Coefficients $A_1$ and $A_2$ can be found out by Solving the System of Linear Equations given by equations (8) and (10) above as follows
      
      $A_1C_2 + A_2C_1=E$   ...(from equation 8)
      
      $A_1D_2 + A_2D_1=G$   ...(from equation 10)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      $\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}E \\ G\end{bmatrix}$   ...(66)
      
      $\Rightarrow \begin{bmatrix}A_1 \\ A_2\end{bmatrix}=\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}E \\ G\end{bmatrix}$   ...(67)
      
      Also, the Values of Coefficients $B_1$ and $B_2$ can be found out by Solving the System of Linear Equations given by equations (9) and (11) above as follows
      
      $B_1C_2 + B_2C_1=F$   ...(from equation 9)
      
      $B_1D_2 + B_2D_1=H$   ...(from equation 11)
      
      The above equations can be given in form of a Matrix Equation as follows
      
      $\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}\begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}F \\ H\end{bmatrix}$   ...(68)
      
      $\Rightarrow \begin{bmatrix}B_1 \\ B_2\end{bmatrix}=\begin{bmatrix}C_2 & C_1 \\ D_2 & D_1\end{bmatrix}^{-1}\begin{bmatrix}F \\ H\end{bmatrix}$   ...(69)
      
      Otherwise, if Values of Both $D_1$ and $D_2$ are Zero, a Quadratic Equation can be formed by using the Values of Coefficients $E$ and $A$ in equations (58) and (54) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Ex + A=0$   ...(70)
      
      $\Rightarrow x^2 - (A_1 + A_2)x + A_1A_2=0$   ...(71)
      
      The Values of Coefficients $A_1$ and $A_2$ can be found out Finding the Roots of Quadratic Equation (70) above.
      
      Similarly, a Quadratic Equation can be formed by using the Values of Coefficients $F$ and $B$ in equations (59) and (55) (which give the Sum of the Roots and Product of the Roots of the Quadratic Equation respectively) as follows
      
      $x^2 - Fx + B=0$   ...(72)
      
      $\Rightarrow x^2 - (B_1 + B_2)x + B_1B_2=0$   ...(73)
      
      The Values of Coefficients $B_1$ and $B_2$ can be found out Finding the Roots of Quadratic Equation (72) above.
      
      The values of $A_1, B_1$ and $A_2, B_2$ can be matched and verified by evaluating equation (57) above.
   5. If the Coefficients of Both $x^2$ ($A$), $y^2$ ($B$) and $z^2$ ($C$) in the equation (1) are Zero and Coefficient of $xy$ ($D$) in the equation (1) is Not Zero, then the Coefficient of $xy$ ($D$) in the Normalized version of equation (1) becomes $1$. Under such condition, one of the following 2 things can be done
      
      Either we can set $A_1=B_2=1$ and $A_2=B_1=0$. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= 0$   ...(74)
      
      $B= 0$   ...(75)
      
      $C= 0$   ...(76)
      
      $D= 1$   ...(77)
      
      $E= C_2$   ...(78)
      
      $F= C_1$   ...(79)
      
      $G= D_2$   ...(80)
      
      $H= D_1$   ...(81)
      
      $I= C_1D_2+C_2D_1$   ...(82)
      
      $K= D_1D_2$   ...(83)
      
      Under such condition, the Value of $C_1=F$, $C_2=E$, $D_1=H$ and $D_2=G$ as given by equations (79), (78), (81) and (80) above respectively.
      
      Otherwize, we can set $A_1=B_2=0$ and $A_2=B_1=1$. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= 0$   ...(74)
      
      $B= 0$   ...(75)
      
      $C= 0$   ...(76)
      
      $D= 1$   ...(77)
      
      $E= C_1$   ...(78)
      
      $F= C_2$   ...(79)
      
      $G= D_1$   ...(80)
      
      $H= D_2$   ...(81)
      
      $I= C_1D_2+C_2_D_1$   ...(82)
      
      $K= D_1D_2$   ...(83)
      
      Under such condition, the Value of $C_1=E$, $C_2=F$, $D_1=G$ and $D_2=H$ as given by equations (78), (79), (80) and (81) above respectively.
   6. If the Coefficients of Both $x^2$ ($A$), $y^2$ ($B$), $z^2$ ($C$) and $xy$ ($D$) in the equation (1) are Zero and Coefficient of $xz$ ($E$) in the equation (1) is Not Zero, then the Coefficient of $xz$ ($E$) in the Normalized version of equation (1) becomes $1$. Under such condition, one of the following 2 things can be done
      
      Either we can set $A_1=C_2=1$ and $A_2=C_1=0$. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= 0$   ...(84)
      
      $B= 0$   ...(85)
      
      $C= 0$   ...(86)
      
      $D= B_2$   ...(87)
      
      $E= 1$   ...(88)
      
      $F= B_1$   ...(89)
      
      $G= D_2$   ...(90)
      
      $H= B_1D_2+B_2D_1$   ...(91)
      
      $I= D_1$   ...(92)
      
      $K= D_1D_2$   ...(93)
      
      Under such condition, the Value of $B_1=F$, $B_2=D$, $D_1=I$ and $D_2=G$ as given by equations (89), (87), (92) and (90) above respectively.
      
      Otherwize, we can set $A_1=C_2=0$ and $A_2=C_1=1$. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= 0$   ...(94)
      
      $B= 0$   ...(95)
      
      $C= 0$   ...(96)
      
      $D= B_1$   ...(97)
      
      $E= 1$   ...(98)
      
      $F= B_2$   ...(99)
      
      $G= D_1$   ...(100)
      
      $H= B_1D_2+B_2D_1$   ...(101)
      
      $I= D_2$   ...(102)
      
      $K= D_1D_2$   ...(103)
      
      Under such condition, the Value of $B_1=D$, $B_2=F$, $D_1=G$ and $D_2=I$ as given by equations (97), (99), (100) and (102) above respectively.
      
      
   7. If the Coefficients of Both $x^2$ ($A$), $y^2$ ($B$), $z^2$ ($C$), $xy$ ($D$) and $xz$ ($E$) in the equation (1) are Zero and Coefficient of $yz$ ($F$) in the equation (1) is Not Zero, then the Coefficient of $yz$ ($F$) in the Normalized version of equation (1) becomes $1$. Under such condition, one of the following 2 things can be done
      
      Either we can set $B_1=C_2=1$ and $B_2=C_1=0$. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= 0$   ...(104)
      
      $B= 0$   ...(105)
      
      $C= 0$   ...(106)
      
      $D= A_2$   ...(107)
      
      $E= A_1$   ...(108)
      
      $F= 1$   ...(109)
      
      $G= A_1D_2 + A_2D_1$   ...(110)
      
      $H= D_2$   ...(111)
      
      $I= D_1$   ...(112)
      
      $K= D_1D_2$   ...(113)
      
      Under such condition, the Value of $A_1=E$, $A_2=D$, $D_1=I$ and $D_2=H$ as given by equations (108), (107), (112) and (111) above respectively.
      
      Otherwize, we can set $B_1=C_2=0$ and $B_2=C_1=1$. Accordingly, the Coefficients and the Constant of the Normalized version of equation (1) above can be given in terms of Coefficients and the Constants of the Linear Equations as follows
      
      $A= 0$   ...(114)
      
      $B= 0$   ...(115)
      
      $C= 0$   ...(116)
      
      $D= A_1$   ...(117)
      
      $E= A_2$   ...(118)
      
      $F= 1$   ...(119)
      
      $G= A_1D_2 + A_2D_1$   ...(120)
      
      $H= D_1$   ...(121)
      
      $I= D_2$   ...(122)
      
      $K= D_1D_2$   ...(123)
      
      Under such condition, the Value of $A_1=D$, $A_2=E$, $D_1=H$ and $D_2=I$ as given by equations (117), (118), (121) and (122) above respectively.