You now have three points that define a plane.

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SUMMARY

The discussion focuses on finding the equation of a plane that passes through the line of intersection of the planes defined by the equations 4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0, while also passing through the point A(1, -3, 2). Participants emphasize the importance of identifying two points on the line of intersection and ensuring that point A is not collinear with those points. The solution involves using the coordinates of these three non-collinear points to derive the equation of the desired plane.

PREREQUISITES
  • Understanding of plane equations in three-dimensional space
  • Knowledge of finding the line of intersection between two planes
  • Ability to determine the equation of a plane given three non-collinear points
  • Familiarity with vector and coordinate geometry concepts
NEXT STEPS
  • Learn how to find the line of intersection of two planes in three-dimensional space
  • Study methods for deriving the equation of a plane from three points
  • Explore the concept of collinearity in three-dimensional geometry
  • Practice solving similar problems involving planes and points in 3D space
USEFUL FOR

Students studying geometry, particularly those focusing on three-dimensional space, as well as educators and tutors looking for examples of plane equations and intersection problems.

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Homework Statement



Find the equation of the plane that passes through the line of intersection of the planes 4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0 and passes through A(1, -3, 2).

Homework Equations



N/A

The Attempt at a Solution



I have no clue on how to start this question, and was hoping that someone could offer some tips to get me started on the problem. I know how to find the line of intersection between two planes, but am unsure on how to find the solution to this question..

Thanks!
 
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You can easily get two points on the line of intersection and you have a third point given. Do you know how to get a plane through 3 non-collinear points?
 
spoc21 said:

Homework Statement



Find the equation of the plane that passes through the line of intersection of the planes 4x - 3y - z - 1 = 0 and 2x + 4y + z - 5 = 0 and passes through A(1, -3, 2).

Homework Equations



N/A

The Attempt at a Solution



I have no clue on how to start this question, and was hoping that someone could offer some tips to get me started on the problem. I know how to find the line of intersection between two planes, but am unsure on how to find the solution to this question..

Thanks!

If you know three points on a plane, and the points aren't all on the same line, you can find the plane that contains the three points, right?

You're given a point. Can you find two points that are on the line of intersection of the two planes? You should satisfy yourself that the given point is not on this line of intersection.
 

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