Homogeneous linear equations geometrically

Click For Summary
SUMMARY

A homogeneous linear equation in three variables, expressed as ax + by + cz = 0, represents a geometric plane in three-dimensional space. When all coefficients a, b, and c are zero, the solution set encompasses all points in space. If any of the coefficients are non-zero, the equation can be manipulated to express one variable in terms of the others, confirming that the solutions form a plane in the three-dimensional coordinate system.

PREREQUISITES
  • Understanding of linear equations and their forms
  • Familiarity with three-dimensional coordinate geometry
  • Basic algebraic manipulation skills
  • Knowledge of geometric interpretations of equations
NEXT STEPS
  • Explore the geometric representation of non-homogeneous linear equations
  • Study the implications of varying coefficients in linear equations
  • Learn about the intersection of planes in three-dimensional space
  • Investigate applications of homogeneous linear equations in computer graphics
USEFUL FOR

Students of mathematics, educators teaching geometry, and professionals in fields requiring spatial analysis, such as computer graphics and engineering.

phymatter
Messages
131
Reaction score
0
what does a homogeneous linear equation in 3 variables represent geometrically ?
 
Physics news on Phys.org
A plane
 
A homogeneous linear equation is of the form ax+ by+ cz= 0 for numbers a, b, c, d. If all of a, b, c, d are 0, that is true for all (x, y, z) so the solution set is all space.

If any of a, b, c, is not 0, we can solve for that variable:
x= (-by- cz)/a
y= (-ax- cz)/b
z= (-ax- by)/c
which are planes,
 
Last edited by a moderator:

Similar threads

Undergrad Gaussian elimination for homogeneous linear systems
  • · Replies 26 ·
Replies
26
Views
5K
Undergrad Dimension and solution to matrix-vector product
  • · Replies 4 ·
Replies
4
Views
3K
High School What unique contributions to math did linear algebra make?
  • · Replies 19 ·
Replies
19
Views
4K
Graduate linear programming, polyhedron and extreme points
  • · Replies 3 ·
Replies
3
Views
1K
MHB Homogeneous, underdetermined equation system
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Solving matrix commutator equations?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad "Rationale" for Homogeneous vs. Nonhomogeneous Differential Equations?
  • · Replies 10 ·
Replies
10
Views
5K
MHB System of linear equations, all possible outcomes
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Geometric intuition of a rank formula
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Gaussian elimination for a singular square matrix
  • · Replies 14 ·
Replies
14
Views
2K