Line-Plane Intersection
In analytic geometry, the intersection of a line and a plane can be the empty set, a point or a line:
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No Intersection
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Point Intersection
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Line Intersection
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How to find the relationship between a line and a plane.
If the line is
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A1x+B1y+C1z+D1=0
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A2x+B2y+C2z+D2=0
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and the plane is
Form a system with the equations and calculate the ranks.
r = rank of the coefficient matrix
r'= rank of the augmented matrix
The relationship between the line and the plane can be described as follow:
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Case 1. Point intersection
r=3 and r'=3
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Case 2. No Intersection
r=2 and r'=3
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Case 3. Line Intersection
r=2 and r'=2
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State the relationship between the line:
and the plane
Solution:
Form the system of equations and calculate the ranks.
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r=3 |
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r'=3 |
Point Intersection.
State the relationship between the line:
and the plane
Solution:
Form the system of equations and calculate the ranks.
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=0 |
r=2 |
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r'=3 |
The line and plane are parallel. There is no intersection.
State the relationship between the plane
