The line L1 passes through the pt A, whose position vector is i-j-5k

and is parallel to the vector i-j-4k. The line L2 passes through the

point B, whose position vector is 2i-9j-14k and is parallel to the vector 2i+5j+6k. The point P on L1 and

the

point Q on L2 are such that PQ is perpendicular to both L1 and L2.

a) Find the length of PQ

I know the vector line of L1 and L2 is L1=i-j-5k+ t(i-j-4k)

L2=2i-9j-14k+s(2i+5j+6k)

The line PQ is perpendicular to both L1 and L2 so I know the common

perpendicular vector.

(1,-1,-4) and (2,5,6) gives a directonal vector of (14i-14j+7k) or

(2,-2,1)

As the distance is perpendicular it must be the minimum? But as each line has a different parameter , s and t, I can't put them into the straight line equation and differentiate.

The book gives an answer of 3, which is the smallest distance I was able to find between the two lines by using some trial and error with different values of s and t.

Thanks