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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 />

4


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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