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              Linda Chuang                                                         September 24 , 2009
              Math 11




              Five different situations of how three planes can be arranged in a
              three-dimensional graph are demonstrated with explanations and diagrams
              below. In each diagram, the red axis represents the x-axis, the blue one
              represents the y-axis, and the green represents the z-axis. In each diagram,
              the equations of the planes can be determined by the coordinates they pass
              through. Then, the equations will be solved to determine the solution to the
              graph. Each diagram is shown with 3 different angles.


              1. First Situation: Three planes intersecting at one point.

                                                               The blue plane passes through the points
(0, 10, 10)                                  (10, 10, 0)       (10,10,0), (10,0,0), (0,10,10), and (0,0,10).


                                                                  1.
                                                                  2.
                                                                  3.
                                                                  4.
     (0, 0, 10)
                                                               One solution:
                                                  (10, 0, 0)
                                                               Therefore, the equation of the blue plane is:




   The yellow plane passes through the points
                                                                                                         (0, 5, 0)
   (0,5,0), (0,5,10), (10,5,10), and (10,5,0).


       1.
       2.
       3.                                                        (10, 5, 0)
       4.


    One solution:                                 .                                                      (0, 5, 10)
    Therefore, the equation of the yellow plane is:
                                                                          (10, 5, 10)
                                                                                                        th
                     Linda Chuang                                                        September 24 , 2009
                     Math 11
(0, 10, 0)                                        (10, 10, 10)        The pink plane passes through the points
                                                                      (0,0,0), (10,0,10), (0,10,0), and (10,10,10).


                                                                         1.
                                                                         2.
                                                                         3.
                                                                         4.


         (0, 0, 0)
                                                        (10, 0, 10)   One solution:                            . Therefore,
                                                                      the equation of the pink plane is:


                     Solving the system:


                    1.
                                                                                         ,          ,
                    2.
                                                                                      The is one solution, which is
                                                                                      (5, 5, 5). The three planes
                    3.
                                                                                      intersect at this point.




                     2. Second Situation: Three planes intersecting along one line.

                             (10, 10, 10)
                                                        (10, 10, 0)   The pink plane passes through the points
                                                                      (0,0,0), (10,10,0), (10,10,10), and (0,0,10).
                                                                      .
                                                                         1.
                                                                         2.
                                                                         3.
                                                                         4.


                                                                      One solution:
                                                                      Therefore, the equation of the pink plane is:
       (0, 0, 10)

                                            (0, 0, 0)
                                                                                           th
         Linda Chuang                                                       September 24 , 2009
         Math 11



The yellow plane passes through the points                                                 (0, 10, 10)

(0,10,0), (0,10,10), (10,0,0), and (10,0,10).      (0, 10, 0)


   1.
   2.
   3.
   4.

One solution:                                 .
Therefore, the equation of the yellow plane is:                                                          (10, 0, 10)

                                                                              (10, 0, 0)




    (5, 10, 10)                (5, 10, 0)         The blue plane passes through the points (5, 10,
                                                  10), (5, 10, 0), (5, 0, 10), and (5, 0, 0).


                                                      1.
                                                      2.
                                                      3.
                                                      4.

                                                  One solution:                                           .
                                                  Therefore, the equation of this plane is:

                  (5, 0, 10)   (5, 0, 0)



         Solving the system:


    1.                                                             ,           ,
                                                            ** Z can be any value. As long as the x and
    2.
                                                           y are 5, any value of z satisfies the equation.

    3.
                                                                There are infinite solutions
                                                                to this system of equations.
                                                                                                         th
                    Linda Chuang                                                            September 24 , 2009
                    Math 11




                    3. Third Situation: Two parallel planes and one intersecting
                    plane.

                                                                                                       (10, 10, 10)
       The blue plane passes through the points
       (0,0,0), (10,10,0), (10,10,10), and (0,0,10).
       .
             1.
                                                                                                                            (10, 10, 0)
             2.
             3.
             4.


        One solution:
        Therefore, the equation of the blue plane is:
                                                                         (0, 0, 10)

                                                                                                              (0, 0, 0)


                                                                      The pink plane passes through the points (0,8,10),
            (0, 8, 10)
                                                                      (0,8,0), (10,6,0), and (10,6,10).


                                                                          1.
                                                                          2.
                                                                          3.
                                                        (10, 6, 10)
                                                                          4.
(0, 8, 0)

                                                                      One solution:                               . Therefore, the
                                                  (10, 6, 0)
                                                                      equation of the pink plane is:



  The pink plane passes through the points
  (0,4,0), (10,2,0), (10,2,10), and (0,4,10).
       1.                                                                                                       (0, 4, 0)

       2.
       3.
       4.                                                        (0, 4, 10)


    One solution:
        Therefore, the equation of the pink plane is:
                                                                                                     th
                   Linda Chuang                                                          September 24 , 2009
                   Math 11

                   Solving the system:


                   1.   –                                                             There are no solutions to
                                                                                      this system of equation.
                   2.                                                                 This means that the three
                                                                                      planes do not intersect at
                   3.                                                                 any point.



                   4. Fourth Situation: Three planes intersecting along three
                   different (parallel) lines.
                                                                The pink plane passes through the points
                                                                (0,0,0), (0,10,10), (10,0,0), and (10,10,10).
                                        (10, 10, 10)
(0, 10, 10)
                                                                   1.
                                                                   2.
                                                                   3.
                                                                   4.


                                                                One solution:                             . Therefore,
                                                                the equation of the pink plane is:

                                                   (10, 0, 0)
                            (0, 0, 0)


                                                                                  (10, 10, 0)                      (0, 10, 0)
         The blue plane passes through the points
         (10,10,0), (0,10,0), (0,0,10), and (10,0,10).


              1.
              2.
              3.
              4.


          One solution:                                    .            (10, 0, 10)
          Therefore, the equation of the blue plane is:                                               (0, 0, 10)
                                                                                                   th
                   Linda Chuang                                                       September 24 , 2009
                   Math 11
                                                              The yellow plane passes through the points
                                                              (10,2,0), (0,2,0), (10,2,10), and (0,2,10).


                                                                 1.
                                                                 2.
                                                (10, 2, 0)
(10, 2, 10)
                                                                 3.
                                                                 4.

                                                  (0, 2, 0)
                                                              One solution:
      (0, 2, 10)
                                                              Therefore, the equation of the blue plane is:




                   Solving the system:


              1.

                                                                              There are no solutions.
              2.
                                                                              The three planes do not
                                                                              intersect.
              3.




                   5. Fifth Situation: Three parallel planes.

                                                                 The yellow plane passes through the points
                                                                 (10,9,10), (10,9,0), (0,8,10), and (0,8,0).
                                  (10, 9, 10)
                                                                      1.
                                                                      2.

                                                (10, 9, 0)            3.
                                                                      4.
(0, 8, 10)

                                                (0, 8, 0)         One solution:                                     .
                                                                  Therefore, the equation of the yellow plane is:
                                                                                                     th
               Linda Chuang                                                              September 24 , 2009
               Math 11




     The pink plane passes through the points (10,6,10),
     (0,5,10), (10,6,0), and (0,5,0).


          1.                                                                                                   (10, 6, 10)

          2.                                                      (0, 5, 10)

          3.                                                                                                           (10, 6, 0)

          4.


      One solution:                                      .               (0, 5, 0)
      Therefore, the equation of the pink plane is:



                                                                   The blue plane passes through the points
                                       (0, 2, 10)
                                                                   (0,2,0), (0,2,10), (10,3,0), and (10,3,10).


                                                                        1.
                                                                        2.
                                                                        3.
                                                                        4.


                                                      (0, 2, 0)      One solution:
(10, 3, 10)
                                                                     Therefore, the equation of the pink plane is:

                   (10, 3, 0)


               Solving the system:


     1.


     2.
                                                                                     .
                                                                                         There are no solutions.
                                                                                         The three planes do not
     3. –                                                                                intersect.

				
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