# Find closest points between lines

1. May 26, 2010

### ThankYou

1. The problem statement, all variables and given/known data
I have two lines :
a,u,b,v are vectors.

$$A=\left\{a+s*u|s \in R \right\} B = \left\{b+t * v|t \in R \right\}$$

The two lines does not touch each other (does not meet)
I need to find the closest point between the lines.

2. Relevant equations

3. The attempt at a solution

I know several ways, But all of them are giving me unbelivable long functions..
There must be a short way.
One options it to build a vector between two random points in the lines and then the scalar multipltion of them need to give me 0 .
a,u,b,v are vectors.
$$(b+t*v-a-s) \bullet v = 0$$
$$(b+t*v-a-s) \bullet u = 0$$
But as I said I tried to solved it and it got to be very very very long and I always made errors...

Second way it to find $$u \times v$$ this is a vector that is vertical to both lines so if I need to fins the solution of :
$$b+t*v+q(u \times v) = a+s*u$$

2. May 26, 2010

### vela

Staff Emeritus
Let me rewrite your last equation a bit.

$$q(\textbf{u} \times \textbf{v}) = (\textbf{a}+s\textbf{u})-(\textbf{b}+t\textbf{v})$$

The RHS corresponds to the vector beginning on a point on B and ending on a point on A. Now try taking the dot product of both sides with $\textbf{u} \times \textbf{v}$. What geometrically does that correspond to?

3. May 27, 2010

### ThankYou

English is not my first language and
Sadly I am not sure I understand what you mean in "try taking the dot product of both sides with $\textbf{u} \times \textbf{v}$"
Do you mean that I need to build three equation :
First we know that $\textbf{u} \times \textbf{v} = (u_{2}v_{3}-v_{2}u_{3} ,-u_{1}v_{3}+v_{1}u_{3} , u_{2}v_{1}-v_{2}u_{1} )$
After we found the vector we can build 3 equations .. Is this is what you meant I need to do>?, $q(u_{2}v_{3}-v_{2}u_{3}) = a_{1}+s*u_{1}-b_{1}-t*v_{1}$<-- Something like this? this is the first equation
Because I did it and it didnt really gave me anything

4. May 27, 2010

### vela

Staff Emeritus
No, that's not what I meant. "Dot product" is another way of saying "scalar product," so I was saying you should do this:

$$q(\textbf{u} \times \textbf{v})\cdot(\textbf{u} \times \textbf{v}) = [(\textbf{a}+s\textbf{u})-(\textbf{b}+t\textbf{v})]\cdot(\textbf{u} \times \textbf{v})$$