Question

How do I find the Cartesian equation of a plane containing two given lines?

Answer 1

Given two lines, they define a plane only if they are:
parallels non coincident or non coincident intersecting.

Explanation:

Given two lines, they define a plane only if they are:
parallels non coincident or non coincident intersecting. If they are parallels, taking a point in one of them and the support of the other we can define a plane. If they intersect, with the normal to both directions and their intersection point, a plane can also be constructed.

`1)` Parallel

`r_1->p = p_1 + lambda_1 vec v`
`r_2->p = p_2 + lambda_2 vec v`

`p_1^0 = p_1 + lambda_1^0 vec v`
`p_2^0 = p_2 + lambda_2^0 vec v`

`vec w = vec v xx (p_1^0-p_2^0) =vec v xx (p_1-p_2)`

the plane equation is given by

`Pi_1-><< vec w, (p-p_1)>> = << vec w, (p-p_2)>> = 0`

here `p = {x,y,z}`

`2)` Intersecting

Solve for the intersection point `p_0`

`p_0 = p_1 + lambda_1 vec v_1 = p_2 + lambda_2 vec v_2`

Once obtained `p_0` the plane is built as follows

`vec w = vec v_1 xx vec v_2`

`Pi_2 -> << vec w, (p-p_0) >> = 0`