Finding the equation of a plane in 3-space

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Discussion Overview

The discussion revolves around finding the equation of a plane in 3-space given specific conditions, such as a point and a parametric equation of a line contained within the plane. Participants explore different scenarios, including cases where the point is on the line or where the plane intersects with other planes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks how to find the equation of a plane with only a point and a line's parametric equation, noting the challenge of having two equations for three unknowns.
  • Another participant suggests creating two vectors from the point to the line if the point is not on the line.
  • It is noted that if the point is on the line, there are infinitely many planes that can be formed.
  • Questions arise about how the two vectors can be used to find the plane, with a focus on the relationship between normal vectors and plane equations.
  • Participants discuss the operation that produces a normal vector from two non-parallel vectors, with one suggesting multiplication as a method to find the normal vector.
  • A participant introduces a new scenario where the plane contains a line that intersects two other planes, asking how to proceed with given equations of those planes and a point on the desired plane.
  • Another participant emphasizes the need for three non-collinear points in the plane to determine its equation, suggesting that the intersection line between the two given planes can be found first.
  • Further clarification is sought regarding the process of finding the line of intersection between two planes and how to utilize that information to find the equation of the desired plane.

Areas of Agreement / Disagreement

Participants express various viewpoints on how to approach the problem, with no clear consensus on the best method to find the plane's equation under different conditions. Some agree on the necessity of finding a normal vector, while others explore different scenarios and their implications.

Contextual Notes

Participants mention the challenge of having three unknowns when working with the equations of two planes, indicating that assumptions about the relationships between the planes and the points involved may affect the approach taken.

HATE-VECTORS
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How do you find an equation of a plane in 3-space when you are only given one point and the parametric equation of a line which is contained within the plane? I tried making an augmented matrix but only have 2 equations for 3 unknowns!
 
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If the point is not on the line, you should easily be able to create two vectors from the point to the line.
 
Yes. If the point is on the line, then there are an infinite number of such planes.

If the point is not on the line, choose any two points on the line and construct the vectors from each to the given point.
 
And then? How do those two vectors help you find the plane?
 
HATE-VECTORS said:
And then? How do those two vectors help you find the plane?

Do you know how to find a normal vector, and how it relates to finding an equation for the plane?
 
I know how it relates to finding the equation. My problem is that I have no clue how to find it:-)
 
HATE-VECTORS said:
I know how it relates to finding the equation. My problem is that I have no clue how to find it:-)

If you have two vectors that are not parallel, do you know an operation that always produces a third vector that is perpendicular to both of them?
 
U multiply them together! and then that gives you the normal cause its perp to the plane as well! Thank you!
 
ok. I get that now. What happens if Instead of being given the equation of a line I am told that the plane contains a line that intersects two other planes. And then I am given just the equation of the two other planes and a point on the plane that I am looking for?
 
  • #10
HATE-VECTORS said:
ok. I get that now. What happens if Instead of being given the equation of a line I am told that the plane contains a line that intersects two other planes. And then I am given just the equation of the two other planes and a point on the plane that I am looking for?

It depends on what information is given. Do you have enough information to determine the two points of intersection between the line and the other two planes?

In general, all you need are three points in the plane of interest, where not all of them are on the same line. Once you have them, you can proceed as before.
 
  • #11
I am not given the equation of a line at all. I am told that two planes (of which the equations are given) intersect at a line. That line passes through another plane. I am given a point in the plane and told to find it's equation> now from what i understand i can't do much with the equations of two planes cause there are three unknown... unl;ess- am i ment to solve in terms of the unkown and use that to choose any two random points and then make vectors to the given point and do what you just told me to do??
 
  • #12
HATE-VECTORS said:
I am not given the equation of a line at all. I am told that two planes (of which the equations are given) intersect at a line. That line passes through another plane. I am given a point in the plane and told to find it's equation> now from what i understand i can't do much with the equations of two planes cause there are three unknown... unl;ess- am i ment to solve in terms of the unkown and use that to choose any two random points and then make vectors to the given point and do what you just told me to do??

You should be able to find the equation of the line of intersection between the two given planes. Once you have done this, you can solve the rest of the problem exactly as before, as long as the given point doesn't lie on the line.

If you're not sure how to find the line of intersection, then show us the given equations for the planes and where you got stuck.
 

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