Intersection of Planes

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Homework Statement


Determine the scalar equation of the plane that contains the line of the intersection of the planes x+y+z=4 and y+z=2, if the plane is two units from the origin.


Homework Equations


direction of intersecting line is M = N1 × N1


The Attempt at a Solution


Let y= 0, find a POI of two planes
x=2, z=2
Therefore one POI is (2,0,2)
Direction of Line is M = N1 × N1 = [1,1,1] × [0,1,1]


Would there not be multiple solutions to this problem because there are multiple of planes that can be rotated around the line of intersection that are all two units from the origin?
 

Answers and Replies

  • #2
LCKurtz
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Homework Statement


Determine the scalar equation of the plane that contains the line of the intersection of the planes x+y+z=4 and y+z=2, if the plane is two units from the origin.


Homework Equations


direction of intersecting line is M = N1 × N1


The Attempt at a Solution


Let y= 0, find a POI of two planes
x=2, z=2
Therefore one POI is (2,0,2)
Direction of Line is M = N1 × N1 = [1,1,1] × [0,1,1]


Would there not be multiple solutions to this problem because there are multiple of planes that can be rotated around the line of intersection that are all two units from the origin?
If by "multiple of planes" you mean as many as two I think I would agree. They would have to be tangent to a sphere of radius two around the origin.
 
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How would I find the scalar equations of those planes?
 
  • #4
LCKurtz
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760

Homework Statement


Determine the scalar equation of the plane that contains the line of the intersection of the planes x+y+z=4 and y+z=2, if the plane is two units from the origin.


Homework Equations


direction of intersecting line is M = N1 × N1


The Attempt at a Solution


Let y= 0, find a POI of two planes
x=2, z=2
Therefore one POI is (2,0,2)
Direction of Line is M = N1 × N1 = [1,1,1] × [0,1,1]
So M = ?? Don't expect us to do the work for you.

If by "multiple of planes" you mean as many as two I think I would agree. They would have to be tangent to a sphere of radius two around the origin.
How would I find the scalar equations of those planes?
Well, once you know M, you know the normal vector to those planes must be perpendicular to M. So what must the normals look like? Once you know that you can use the normals and the point (2,0,2) to write their equations.
 

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