Homogeneous linear equations geometrically

Thread starter phymatter
Start date Feb 2, 2011
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Homogeneous Linear Linear equations

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A homogeneous linear equation in three variables, expressed as ax + by + cz = 0, represents a geometric plane in three-dimensional space. If all coefficients a, b, and c are zero, the solution set encompasses all points in space. When at least one coefficient is non-zero, the equation defines a specific plane determined by the relationships among the variables. The solutions can be rearranged to express one variable in terms of the others, illustrating the plane's orientation. Understanding these equations is crucial for visualizing linear relationships in higher dimensions.

Feb 2, 2011

phymatter

what does a homogeneous linear equation in 3 variables represent geometrically ?

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Feb 2, 2011

!^--_--^!

A plane

Feb 2, 2011

HallsofIvy

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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: Feb 2, 2011

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