Line of Intersection of 3 planes

In summary, the given systems of equations do not have a solution and therefore have no point of intersection. This can be determined by checking for coplanarity and parallelism of the vectors, as well as finding the common line between any two of the given planes and checking if it lies on the third plane.
  • #1
mikee
30
0

Homework Statement

Solve the following systems and interpret the result geometrically
3x + 4y + 5z - 18 = 0
2x - y + 8z - 13 = 0
-x + 17y + 25z + 11 = 0




Homework Equations





The Attempt at a Solution

I've been working on this problem for a while, The first thing i did was find if the vectors are coplanar by V1 . V2xV3 and they are not and then i checked there to see if they were parralel and neither are parralel to the other, so therefore there should be a point of intersection right? But i tryed solve it over and over again but i couldn't come up with a solution, that is could not find x,y and z that satifies all 3 equations which therefore means there is no solution and no point of intersection? am i doing somthing wrong or is this question faulty.
 
Physics news on Phys.org
  • #2
Hi there mikee! :smile:

Three planes can intersect in a line …

take any two of the given planes, and fine their common line.

Then check to see if that line happens to be on the third plane. :smile:
 

1. What is the definition of a line of intersection of 3 planes?

A line of intersection of 3 planes is the line that exists where all 3 planes intersect. This line is formed by the points that are common to all 3 planes.

2. How can I determine if 3 planes intersect at a single point or form a line of intersection?

If the equations of the 3 planes have a single unique solution, then they intersect at a single point. If the equations have infinite solutions, then the planes intersect at a line. This can be determined by solving the system of equations using methods such as substitution or elimination.

3. Can a line of intersection of 3 planes be parallel to any of the planes?

No, a line of intersection of 3 planes cannot be parallel to any of the planes. This is because if a line is parallel to a plane, it would not intersect the plane at any point, which contradicts the definition of a line of intersection.

4. How many points are needed to uniquely define a line of intersection of 3 planes?

A line of intersection of 3 planes can be uniquely defined by two points. These two points are the intersection of the line with any two of the three planes.

5. Can 3 planes intersect at more than one line of intersection?

Yes, it is possible for 3 planes to intersect at more than one line of intersection. This occurs when the planes are not all parallel to each other, and the lines of intersection are not parallel to each other either.

Similar threads

Finding line where two planes intersect
    • Calculus and Beyond Homework Help
Replies
3
Views
908
Find the vector equation of the line that passes through the point P and intersects with the straight lines R and S
    • Calculus and Beyond Homework Help
Replies
5
Views
943
How do I find a plane that contains two given lines?
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Finding the point on the line of intersection between planes
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Find intersections between sphere and parallel tangent planes
    • Calculus and Beyond Homework Help
Replies
7
Views
2K
The Line of Intersection of Two Planes
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Intersection of line through a plane
    • Calculus and Beyond Homework Help
Replies
2
Views
922
Equation of line intersecting two planes
    • Calculus and Beyond Homework Help
Replies
3
Views
2K
Applying Stokes' Theorem to the part of a Sphere Above a Plane
    • Calculus and Beyond Homework Help
Replies
21
Views
3K
Finding the Line of Intersection for Two Cylinders
    • Calculus and Beyond Homework Help
Replies
16
Views
2K

Hot Threads

Recent Insights

Back
Top