+845 
For a i got r= (-2,3,4) + λ(3,2,-7)
for b so far i wanted to identify a point A on l1 and a point b on l2
i assumed the two lines are parallel as they have the same direction vector
\(A=\begin{pmatrix} -2+3λ\\ 3+2λ\\ 4-7λ \end{pmatrix}\) \(B= \begin{pmatrix} 3\\ 1+2λ\\ -2-7λ \end{pmatrix}\)
used this to get
\(AB= \begin{pmatrix} 5-3λ\\ -2\\ -6 \end{pmatrix}\)
used this and my direction vector to find λ=53/9 via dot product
substituted and calculated the magnitude which gave me 14.26
but my answer should be 3.57
please identify what went wrong, thanks
The equation of the first line can be written as r1 = (-2, 3, 4) + s(3, 2, -7), (as you calculated),
and the equation of the second as r2 = (0, 1, -2) + t(3, 2, -7), s and t arbitrary parameters.
A vector d connecting two points, one on each line, would be given by
d = r2 - r1 = (2, -2, -6) + (t - s)(3, 2 ,-7) = (2, -2, -6) + w(3, 2, -7), w for convenience.
The distance between the two points will be the magnitude of this vector, so what you are looking for is the minimum value of
sqrt{(2 + 3w)^2 + (-2 + 2w)^2 + (-6 -7w)^2} = sqrt{62w^2 + 88w + 44} and that turns out to be 3.5741 (4dp).
Tiggsy
