Embedding a line into a plane

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In summary, to find the equation of a plane containing one line, L1, and parallel to another line, L2, you can use the cross product of the direction vectors of the lines to determine the normal vector of the plane. Then, using a point from either line, you can solve for the remaining coefficient in the equation of the plane.
  • #1
phoneprinter
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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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  • #2
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
 
  • #3
Thank you, but do you think you could give me an example?
 
  • #4
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...
 
  • #5
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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  • #6
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.
 
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  • #7
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 )
 

1. What is the purpose of embedding a line into a plane?

Embedding a line into a plane is a mathematical concept that allows us to represent a line on a 2-dimensional plane. This is useful for visualizing and studying the properties of lines and their intersections with other geometric shapes.

2. How is a line embedded into a plane?

A line can be embedded into a plane by defining a set of coordinates for points on the line and plotting them on the plane. The line can then be extended infinitely in both directions.

3. What is the difference between embedding a line and drawing a line on a plane?

When a line is drawn on a plane, it is limited to the specific points and length that are drawn. However, when a line is embedded into a plane, it extends infinitely in both directions, allowing for a more accurate representation and study of its properties.

4. Can any line be embedded into any plane?

Yes, any line can be embedded into any plane as long as the plane is 2-dimensional and the line is a 1-dimensional object.

5. What are the practical applications of embedding a line into a plane?

Embedding a line into a plane has many practical applications in fields such as engineering, architecture, and computer graphics. It allows for the accurate representation of lines in 2-dimensional designs and simulations.

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