# Thread: Endpoints of shortest distance between Skew lines

1. ## Endpoints of shortest distance between Skew lines

I know the forumla to find the distance between two skew lines in space. But, I need to find out the coordinates on the two lines that represent the shortest distance.

2. Originally Posted by jsmith90210
I know the forumla to find the distance between two skew lines in space. But, I need to find out the coordinates on the two lines that represent the shortest distance.
Let

$\displaystyle L_1:\vec r = \vec a+s\cdot \vec u$
and
$\displaystyle L_2:\vec r = \vec b + t\cdot \vec v$

denote the equations of the two lines. I assume that $\displaystyle L_1\cap L_2=\emptyset$ and that exist $\displaystyle \vec n = \vec u \times \vec v$

To calculate the endpoints of the shortest distance between arbitrary points of the two lines the distance vector must be a multiple of the normal vector:

$\displaystyle (\vec a - \vec b)+s\cdot \vec u - t\cdot \vec v = k\cdot \vec n$

You'll get a system of 3 simultaneous equations. Solve for (s, t, k). Plug in the values of s and t into the appropriate equations to get the endpoints of the distance.

3. I'm sorry, but I'm not sure i understand the form you're using for the equation of a line. I've learned that the equation for a line is like the following:

(2+3d)i + (3+2d)j + (1+4d)k

Where: (2, 3, 1) is a point on the line. And 3i + 2j + 4k would be the vector representing the orientation of the line