Intersection of Plane x=y and Surface in R3

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Start date Jul 3, 2011
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Intersection Plane Surface

In summary, the intersection of a plane and a surface in R3 is the set of points that lie on both the plane and the surface. To find this intersection, one can solve the system of equations formed by the plane and surface. This intersection can be a line, a point, or even an empty set depending on the equations. It is possible for a plane and surface to intersect at more than one point or have no intersection at all. This intersection is a 3D representation of the solution set and allows for visualizing the points that satisfy both equations in 3D space.

Jul 3, 2011

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What does a surface in R3 that intersects plane x = y at a line for every value of x represent?

My first intuition is that it represents a plane because in R3 planes intersect at lines but I feel like there is a counterexample to this.

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Jul 3, 2011

SammyS

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It doesn't need to be s plane.

The surface z = x² intersects the plane y = x for every value of x.

So does x = z³.

FAQ: Intersection of Plane x=y and Surface in R3

1. What is the intersection of a plane and a surface in R3?

The intersection of a plane and a surface in R3 is the set of points that lie on both the plane and the surface. It can be a line, a point, or even an empty set depending on the specific equations of the plane and surface.

2. How do you find the intersection of a plane and a surface in R3?

To find the intersection of a plane and a surface in R3, you can solve the system of equations formed by the equations of the plane and surface. This will give you the coordinates of the points that lie on both the plane and surface, if they exist.

3. Can a plane and a surface in R3 intersect at more than one point?

Yes, a plane and a surface in R3 can intersect at more than one point. This can happen if the plane and surface are not parallel and their equations have multiple solutions when solved together.

4. Is it possible for a plane and a surface in R3 to have no intersection?

Yes, it is possible for a plane and a surface in R3 to have no intersection. This can happen if the plane and surface are parallel and their equations have no solutions when solved together.

5. How does the intersection of a plane and a surface in R3 relate to the concept of a 3D graph?

The intersection of a plane and a surface in R3 is a 3D representation of the solution set of the system of equations formed by the plane and surface. It is a way to visualize the points in 3D space that satisfy both equations simultaneously.