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th 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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