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