How Do You Find a Plane That Contains One Line and Is Parallel to Another?

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Discussion Overview

The discussion revolves around finding the equation of a plane that contains one line (L1) and is parallel to another line (L2) in three-dimensional space. Participants explore various methods and considerations related to this geometric problem, including vector representations and the properties of parallel lines.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests that any vectors perpendicular to the plane will also be perpendicular to the lines within it, hinting at the use of direction vectors to find a normal vector.
  • Another participant emphasizes the need for an example to clarify the initial hints provided.
  • A later reply indicates that the problem is basic and suggests looking into geometry books for foundational understanding, while also acknowledging their own uncertainty about the topic.
  • One participant proposes using the cross product of the direction vectors of the two lines to find the normal vector to the plane, assuming the lines are parallel and coplanar.
  • Another participant raises a critical point about the conditions under which the cross product would be valid, questioning whether the lines are indeed parallel and coplanar or if L1 is in the plane while L2 is parallel to it, noting that this distinction affects the existence of a unique plane.
  • Another response provides a formula for the plane's equation based on the cross product of the direction vectors, but it assumes specific conditions about the lines' relationships.

Areas of Agreement / Disagreement

Participants express differing views on the conditions necessary for finding the plane, particularly regarding the relationship between the lines. There is no consensus on the correct approach, as some participants propose methods based on different assumptions about the lines' configurations.

Contextual Notes

Participants highlight the importance of clarifying whether the lines are parallel and coplanar or if one line is in the plane while the other is parallel to it, as this affects the applicability of various mathematical methods discussed.

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Suppose you have two lines in parametric form that do not intersect. How can you find the equation of a plane containing one line, L1, and that is also parallel to the other line, L2?

Any help would be greatly appreciated, thank you very much.
 
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There are lots of way to solving your problem.
A Hint: any vectors that are perpendicular(?) to the plane will perpendicular to the lines in it.
From any lines' equations, you can take out the dir vectors to compute the ones that are "normal" to them
 
Thank you, but do you think you could give me an example?
 
Sure, but please go search your school library first, it is a very basic 3d problem. It has been years since I last solved 3d problems like what you ask here...


Note: Please do not get me wrong, I said "basic" because your question can be answered in some of the first parts of geometry books...I actually also fogot a lot about this, and what I suggested was just a hint which at least I think or know for sure should be one of many possible ways for you to make a start to retrieving the plane you are trying...
 
Here's one way :

Write the line equations in the vector form. The cross product of the 2 direction vectors gives you the normal (N) to the plane. A point (R) on the plane can be selected from L1. From R and N you can find the equation of the plane.
 
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phoneprinter said:
Suppose you have two lines in parametric form that do not intersect. How can you find the equation of a plane containing one line, L1, and that is also parallel to the other line, L2?

Any help would be greatly appreciated, thank you very much.


i want to make sure: you mean the lines are parallel and coplanar right? NOT L1 is in the plane and the plane is parallel to L2, correct? If you mean the former then the cross product will not work as others have suggested, it will always be a null vector. If you mean the second then there is no unique plane that can be defined.


Assuming you mean the former, that the lines are parallel and coplanar and you want to find the plane containing them, then find the vector connecting a point on L1 to a point L2 (which points are irrelevant) and take the cross product of that vector with the direction vector of the lines (if the lines are parallel they have the same direction vector). this will give you the normal vector to the plane, and you can its components as the coefficients of x, y, and z in the equation of the plane, and solve for d using a point from either line.
 
Last edited:
phoneprinter said:
Suppose you have two lines in parametric form that do not intersect. How can you find the equation of a plane containing one line, L1, and that is also parallel to the other line, L2?

Any help would be greatly appreciated, thank you very much.
If you have two lines in parameter form:

L1 : p(t) = p1 + v1*t
L2 : p(t) = p2 + v2*t

Then the equation of the plane containing L1 and parallel to L2 is given by:

(p-p1)*n = 0

With n = cross_product(v1(x1,y1,z1),v2(x2,y2,z2)) = (y1*z2-z1*y2, z1*x2-x1*z2 , x1*y2-y1*x2 )
 

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