Intersection of Two Planes in R3: Always a Line?

In summary, the intersection of 2 planes is the point or line where two planes intersect in three-dimensional space. To find this intersection, the equations of the planes can be set equal to each other and solved for the variables. Two planes can intersect at a single point, a line, or be parallel and never intersect, depending on their equations. The distance between parallel planes can also be calculated using their equations. The intersection of 2 planes has real-life applications in fields such as engineering, architecture, and physics.
  • #1
mirandasatterley
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0

Homework Statement



True or False: The intersection of two planes in R3 is always a line.

The Attempt at a Solution



I'm pretty sure that this statement is true because two planes can only be parallel, or they must intersect in a line because the are infinate.

But I have no ideas on how to prove this mathematically. Any ideas?
 
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  • #2
Are you familiar with the cross product? If so, consider the cross product of the normals to each plane.

By the way, what if the planes are parallel?
 
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Related to Intersection of Two Planes in R3: Always a Line?

1. What is the definition of "Intersection of 2 planes"?

The intersection of 2 planes refers to the point or line where two planes intersect or overlap in three-dimensional space. This point or line is determined by the equations of the two planes.

2. How do you find the intersection of 2 planes?

To find the intersection of 2 planes, you can use the method of substitution or elimination. First, set the equations of the two planes equal to each other. Then, solve for the variables to determine the coordinates of the point or the equations of the line of intersection.

3. Can two planes intersect at more than one point?

Yes, two planes can intersect at a single point, a line, or they can be parallel and never intersect. The type of intersection depends on the relationship between the equations of the two planes.

4. What does it mean when two planes are parallel?

When two planes are parallel, it means that they never intersect and are always the same distance apart. This distance is known as the distance between the two planes and can be calculated using the equations of the planes.

5. How is the intersection of 2 planes used in real life?

The intersection of 2 planes is used in various fields such as engineering, architecture, and physics. For example, in engineering, the intersection of two planes can be used to determine the angle between two surfaces or the location of a point in space. In architecture, it can be used to design structures with complex angles and intersections. In physics, the intersection of two planes can represent the path of two objects in motion and their collision point.

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