9.4 Intersection of 3 Planes
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What is the simple test we use to determine whether or not the normals are coplanar?
This formula gives us the volume of the parallelepiped formed by the 3 normals. If this product is zero, then the normals all lie on the same plane (coplanar). If we don't get zero, then the normals are not coplanar and we have a unique point of intersection.
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9.4 Intersection of 3 Planes
Case 1: The system has a unique solution. The three planes intersect at only one point. If n1, n2, and n3 are not coplanar, then the planes intersect in a single point.
Example 1: Determine the intersection of the three planes.
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Solve Using Matrices
=
2
0
3
1
3
1
1
­2
2
­ 4
2
­ 7
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Point of Intersection
Solve:
• x ­ y + z = ­2
‚ 2x ­ y ­ 2z = ­9
ƒ 3x + y ­ z = ­2
Step 1: Create two equations „ and … each with an x term of zero.
•
„
…
x ­ y + z = ­2
0x + y ­ 4z = ­5
0x + 4y ­ 4z = 4
­ 2• + ‚
­3• + ƒ
Step 2: Create equation †by eliminating y from equations
„ and …
• x ­ y + z = ­2
„ 0x + y ­ 4z = ­5
† 0x + 0y + 12z = 24
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Using Matrices ­ Thanks Tony!
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1
2
3
­1 1
­1 ­2
1 ­1
­2
­9
­2
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Case 2: The system has an infinite number of solutions described by one parameter, in which case
the three planes intersect in a line.
Example 2: Find the intersection of the planes
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Example 2 Using Matrices
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Now you try:
3x + 2y ­ z = 0
3x ­ 5y +4z = 3
2x ­ y + z = 1
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Using Matrices!
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Case 3: The system has an infinite number of solutions described by two parameters, in which case the three planes are coincident and the solution consists of the coordinates of all points in the plane
Example 3: Describe the intersection of the planes
pg 531 #8abcd,13a
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pg 531 #8abcd,13a
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Case 4: The system has no solutions; that is, it is inconsistent. This will happen if at least two of the planes are parallel and distinct. It will also happen if the three lines of intersection of pairs of planes are parallel; in this case the planes bound an infinite triangular prism
Example 4: Describe the intersection of the planes
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Example 5:
Determine the intersection of the planes.
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Consider the three planes
j
k
l
State the normal vectors for each plane above.
Explain how you would determine if the planes are distinct or coincident? What constant terms in equations k and l would make these equations represent the same plane as equation j.
The three planes are all parallel to one another. The diagram below shows a side view of the planes, which appear as parallel lines on the page. The planes come out of the page towards the viewer. The normal vectors are perpendicular to the planes, and lie flat on the page. The normal vectors are collinear, and also coplanar.
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Intersections of Three Planes
Suppose three distinct planes have normal vectors , n1, n2 , and n3 . To determine if there is a unique point of intersection, calculate • If , the normal vectors are not coplanar. There is a single point of intersection.
• If , the normal vectors are coplanar. There may or may not be points of intersection. If there are any points of intersection then they lie on a line.
Homework: pg 531 #1,3,5, 6, 8,9, 12, 13
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