Determining the cartesian equations

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SUMMARY

The discussion focuses on determining the Cartesian equation of a plane containing two lines, L1 and L2, defined by their parametric equations. The lines are given as L1: r = (4,4,5) + t(5,-4,6) and L2: r = (4,4,5) + s(2,-3,-4). To find the plane's equation, users must utilize the formula Ax + By + Cz + D = 0, where the coefficients A, B, and C represent the components of a normal vector derived from the cross product of the direction vectors of the two lines. Understanding the relationship between the lines and their direction vectors is crucial for solving the problem.

PREREQUISITES
  • Understanding of vector operations, specifically cross products
  • Familiarity with parametric equations of lines
  • Knowledge of Cartesian equations of planes
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study vector cross product calculations to find normal vectors
  • Learn how to derive Cartesian equations from parametric equations
  • Explore examples of planes defined by two lines in 3D space
  • Review the geometric interpretation of lines and planes in three dimensions
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Students studying geometry, particularly those tackling problems involving planes and lines in three-dimensional space, as well as educators looking for teaching resources on vector mathematics.

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Homework Statement



determine the cartesian equation of the plane that contains the following lines:
L1: r= (4,4,5) + t(5,-4,6)
L2: r= (4,4,5) + s(2,-3,-4)

Homework Equations



I kno I'm supposed to use the equation Ax + By + Cz + D. but i don't know how to use it with this type of problem

The Attempt at a Solution



Don't understand it at all :S
 
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Two lines can be contained in a plane if and only if,

a) The lines are parallel, or
b) The lines intersect

Hint: The direction vectors of each line should indicate whether they are parallel or not

Does that help?
 
Last edited:
shawns said:

Homework Statement



determine the cartesian equation of the plane that contains the following lines:
L1: r= (4,4,5) + t(5,-4,6)
L2: r= (4,4,5) + s(2,-3,-4)

Homework Equations



I kno I'm supposed to use the equation Ax + By + Cz + D. but i don't know how to use it with this type of problem
Ax + By + Cz + D is NOT an equation. The equation you're thinking of is Ax + By + Cz + D = 0.
shawns said:

The Attempt at a Solution



Don't understand it at all :S

You are given a point in the plane and two vectors that are in the plane. You need to work with the two vectors to find a third vector that is perpendicular to the plane.
 

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