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On differentiability

Let f:R-->R^n and g:R-->R^n be two differentiable curves with f`(t)=/=0 and g`(t)=/=0 for all t in R. Assume that p=f(s_0) and q=g(t_0) are closer than any other pair of points on the two curves. Prove that p-q is orthogonal to both f`(s_0) and g`(t_0).
Apply this result to find the closest pair of points of the skew straight lines in R^3 defined by f(s)=(s,2s,-s) and g(t)=(t+1,t-2,2t+3).

Any advice on how to approach this?
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RichE
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J.F.N
Let f:R-->R^n and g:R-->R^n be two differentiable curves with f`(t)=/=0 and g`(t)=/=0 for all t in R. Assume that p=f(s_0) and q=g(t_0) are closer than any other pair of points on the two curves. Prove that p-q is orthogonal to both f`(s_0) and g`(t_0).
Apply this result to find the closest pair of points of the skew straight lines in R^3 defined by f(s)=(s,2s,-s) and g(t)=(t+1,t-2,2t+3).

Any advice on how to approach this?


Let D(s,t) = |p(s)-q(t)|^2 = (p(s)-q(t)).(p(s)-q(t))

At its minimum the partial derivatives dD/ds and dD/st will both be zero - i.e.

(p-q).f' =0
(p-q).g'=0

Then solve these simultaneous equations for s and t in the given example.

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