# Embedding a line into a plane

1. May 23, 2004

### phoneprinter

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.

Last edited: May 23, 2004
2. May 23, 2004

### Pattielli

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. May 23, 2004

### phoneprinter

Thank you, but do you think you could give me an example?

4. May 24, 2004

### Pattielli

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. May 24, 2004

### Gokul43201

Staff Emeritus
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.

Last edited: May 24, 2004
6. May 24, 2004

### franznietzsche

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: May 24, 2004
7. May 25, 2004

### Simon666

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 )