Find closest points between lines

[ThankYou]

Homework Statement

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.

Homework Equations

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

 

[vela]

Homework Statement

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.

Homework Equations

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

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?

 

[ThankYou]

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?

Thank you for your respond.
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

 

[vela]

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})