What is the System of Equations for Finding Planes Containing a Given Line?

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Homework Help Overview

The discussion revolves around finding a system of equations that defines planes containing a given line represented in parametric form. The subject area includes geometry and linear algebra, specifically focusing on the relationship between lines and planes in three-dimensional space.

Discussion Character

  • Exploratory, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to derive a system of equations based on a line's coordinates and seeks a second equation to complement the first. Other participants suggest practical methods, such as selecting additional points on the line to generate further equations.

Discussion Status

The discussion is ongoing, with participants exploring different approaches to formulate the necessary equations. Some guidance has been offered regarding the selection of points on the line, but no consensus has been reached on how to proceed with obtaining the second equation.

Contextual Notes

The original poster has provided one equation derived from a point on the line but indicates a lack of information on how to obtain a second equation. There may be constraints related to the requirements of the problem or the specific points chosen from the line.

jdstokes
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Let [itex]\ell[/itex] be the line given by [itex]\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z-3}{4}[/itex]. Write down a system of two equations in the three unknowns [itex]a[/itex], [itex]b[/itex], [itex]c[/itex] whose solutions give all planes [itex]ax + by + cz = 1[/itex] in which [itex]\ell[/itex] lies, and solve the system. I can certainly write that [itex]a -2b + 3c = 1[/itex]. I can't figure out how to obtain the second equation.

Thanks

James.
 
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Just pick two points on the line and plug in for (x,y,z).
 
I picked (1,-2,3) and I only have one equation [itex]a -2b + 3c = 1[/itex].
 
so pick another one...
 

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