# Prove distance between lines

1. May 25, 2010

### ThankYou

1. The problem statement,
all variables and given/known data

There are 4 vectors a,b,u,v in space and {u,v} are linearly independen
The groups A={a+s*u|s E R} B = {b+t * v|t E R}are two lines in space

The Two A , B lines does not meet and the "part/line/segment (I dont know how it called ) " PQ is the smallest part that is ends are $$Q \in B ,P \in A$$ show that the length of PQ is"
$$\frac{|(u \times v)\bullet(a-b)|}{||u \times v||}$$

2. Relevant equations

all calc/..

3. The attempt at a solution
I mange to get to the point of
$$u \times v =(s)$$
and then
$$\frac{L(s)*L(a-b)*cos(w)}{L(s)}=L(a-b)*cos(w)$$
bUT I dont know whats next

2. May 26, 2010

### gabbagabbahey

Huh? How are you managing to get to this point? What do $w$ and $L$ represent?

What is the general formula for the distance between two points $\textbf{r}_1$ and $\textbf{r}_2$? What if the points lie on the lines given in the problem (i.e. $\textbf{r}_1=\textbf{a}+s\textbf{u}$ and $\textbf{r}_2=\textbf{b}+t\textbf{v}$)? How would you go about minimizing that distance?

3. May 26, 2010

### ThankYou

Yes I've managed to do it...
I've done:
$$u \times v$$ and got the vector that is vertical to both u , v
Then I've made a plane that use this vector and use point a, then I've used the forumala of plane distance from point in the first line.
Thank you/