Sec 9.4 The intersection of Three Planes
There are 2 different ways that 3 planes can intersect:
Consistent systems have one solution
Inconsistent systems have more than one solution
or no solution
Jun 6­12:02 PM
Consistent Systems
See diagrams and cases on p521
Case 1: The 3 planes intersect in a single point
Case 2: There are an infinite # of solutions that form a line
a) none of the planes are the same and they meet in a line
b) two of the three planes are the same and they meet in a line
Case 3: There are an infinite # of solutions that form a plane
The three planes are the same plane (coincident)
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We can use a matrix approach or an elimination approach to isolate each variable.
Note, because we found a unique point, we are looking at a Case 1 scenario, where three planes intersect at one point.
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Using technology and a matrix approach we can verify our solution.
http://math.nelson.com/calculusandvectors/web_links.html
Choose the systems of equations solver and resolve the previous example.
Jun 6­11:50 AM
Before we start, we should investigate the normals of each plane to see if any more insight is given to the orientation of the planes.
Operational Approach
Matrix Approach
Because we end up with 0z=0, we have found a solution that has an infinite number of solutions; to accommodate this information we will introduce the parameter "z = t" and back substitute into the two other equations.
Jun 6­12:19 PM
Before we start, we should investigate the normals of each plane to see if any more insight is given to the orientation of the planes.
Operational Approach
Matrix Approach
Because we end up with 0z=0, we have found a solution that has an infinite number of solutions; to accommodate this information we will introduce the parameter "z = t" and back substitute into the two other equations.
Jun 6­12:30 PM
Inconsistent Systems for Three Equations Representing Three Planes
Jun 6­12:12 PM
Inconsistent Systems
See diagrams and cases on p526
Case 1: The three planes meet in 3 distinct parallel lines ­ form a triangular prism
Case 2: Two of the planes are parallel and the third passes through them ­ the solution is 2 distinct lines
Case 3: All three planes are parallel ­ none coincident so no solution.
Case 4: All three planes are parallel but 2 are coincident so no solution
Mar 10­1:21 PM
Before we start, we should investigate the normals of each plane to see if any more insight is given to the orientation of the planes.
Operational Approach
Matrix Approach
Note the difference between an infinite solution scenario, "0z = 0" vs "0z = 3" which is impossible to solve.
From the normal vectors we know that the three planes are not parallel and the solution of the system has returned no solution, we must be looking for a case 1 or triangular prism situation.
Reminder, by crossing the normals of two intersecting planes we can determine the direction vector of the line of intersection. We can find the three direction vectors by performing the following:
Jun 6­12:46 PM
The key indicators for recognizing and inconsistent systems are:
1. the "no solution" of the system of equations
2. watching for planes that are parallel but not coincident.
Jun 6­12:43 PM
Jun 6­1:06 PM
Jun 6­1:06 PM
Consolidation Questions:
page 530 ­ 33 #5, 6, 8­11, 12, 13 (selection)
Jun 6­1:06 PM