1. The direction vectors of the lines are different therefore the lines are not parallel.

2. An attempt to calculate the point of intersection doesn't yield a result therefore the lines are skewed.

3. The cross-product of the direction vectors yield the direction perpendicular to both lines:

4. Now use this direction vector and any arbitrary point on to get the equation of a family of parallel lines:

....... r is considered to be constant with a certain line of the family.

5. One (and only one) member of this family will intersect the line :

6. Solve this system of linear equations for r, s and q.

I've got

7. The 2 points on different lines which have the smallest distance are:

and

8. Take one of the two points to calculate the equation of the line:

My results do look really ugly, so better check my arithmetic. (If they are miraculously correct you are allowed to think two or three ugly words about the author of this question)