Line of Intersection of 3 planes

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SUMMARY

The discussion centers on solving the system of equations representing three planes: 3x + 4y + 5z - 18 = 0, 2x - y + 8z - 13 = 0, and -x + 17y + 25z + 11 = 0. The initial analysis confirmed that the vectors are neither coplanar nor parallel, suggesting a potential intersection. However, the conclusion drawn was that there is no solution, indicating that the planes do not intersect at a single point. The correct approach involves finding the intersection line of any two planes and verifying its presence on the third plane.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically systems of equations.
  • Familiarity with vector operations, including dot and cross products.
  • Knowledge of geometric interpretations of planes in three-dimensional space.
  • Ability to manipulate and solve equations involving multiple variables.
NEXT STEPS
  • Learn how to find the line of intersection of two planes using parametric equations.
  • Study the geometric interpretation of three-dimensional planes and their intersections.
  • Explore the use of matrix methods, such as Gaussian elimination, to solve systems of equations.
  • Investigate the conditions under which three planes can intersect at a single point, a line, or not at all.
USEFUL FOR

Students studying linear algebra, mathematicians interested in geometric interpretations, and educators teaching systems of equations and their applications in three-dimensional space.

mikee
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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.
 
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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:
 

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