Math Help - Distance between 2 parametric lines - Vectors/Lines and Planes

1. Distance between 2 parametric lines - Vectors/Lines and Planes

Hey guys,

I have two lines with two different parametric equations.
And I was asked to find the shortest distance between the two lines.

Suppose the we have two skew lines, $l_1 (t) = P + tD\quad \& \quad l_2 (t) = Q + tE$

What I did was finding D x E, cross product of two directional vectors of both lines, that will give n, the normal vector to both lines.

Now I'm stuck and don't know what to do next so can someone please help me out, it'd be great if you can explain it instead of just formulas, thank you so much

2. Originally Posted by shiiganB
Hey guys,

I have two lines with two different parametric equations.
And I was asked to find the shortest distance between the two lines.

Suppose the we have two skew lines, $l_1 (t) = P + tD\quad \& \quad l_2 (t) = Q + tE$

What I did was finding D x E, cross product of two directional vectors of both lines, that will give n, the normal vector to both lines.

Now I'm stuck and don't know what to do next so can someone please help me out, it'd be great if you can explain it instead of just formulas, thank you so much
1. The first step is OK. Since you are interested in a distance you have to use the unit normal vector $\vec n = \frac{\vec D \times \vec E}{|\vec D \times \vec E|}$

2. Use the dot-product $\vec n \cdot (\vec Q - \vec P)$ to calculate the perpendicular projection of $\vec Q - \vec P$ onto $\vec n$ - and that's exactly the shortest distance between the 2 skew lines.

3. I've attached a sketch. The labeling is different to yours but I'm sure you'll see the similarity to your question.