OK ... if instead of

FindMinimum[dsq[c[t],q],{t,.3,.301}]

we define

f[t_?NumberQ] := dsq[c[t], q];

Then

FindMinimum[f[t], {t, .3,.301}];

works.

I can't say that i understand why it shouldn't work in the original. But I guess

f[t_?NumberQ] 

for the function to be minimized avoids all sorts of stuff I don't understand. It just tells FindMinimum "Just do the numbers". ... and we both understand numbers in the same way.

Well,, thanks for the reply. But why "it treats c[t] as a number,"? when it should know that c[t] is a 2-list, since it was generated that way. As I said, too deep for me. My conclusion: just make the argument of FindMinimum a function that only takes numerical arguments and returns a numerical result, and mysterious "treats as" questions go away...

I think (the Mathematica priests will say) that c is a function whose values are 2-lists. It is not two functions. As i say, this is too subtle for me ...

I wouldn't call myself a priest of Mathematica, and you surely have exposed a pitfall in the system, into which I could have fallen myself. I am just trying to explain the reasons behind the problem. Your symbolic expression c[t] has an InterpolatingFunction object as Head; it only becomes a List when t is a number. When faced with c[t]+{a,b}, the c[t] is not treated as a list. Yes, I suppose that Mathematica could be programmed to check if c[t] could become a list of length 2 when t is numeric, but this would imply messing with addition, and making it slower in many other circumstances (maybe, I am just guessing). So, until Wolfram changes the behaviour of Plus, the creator of a vector-valued InterpolatingFunction must take care of its creature. The same pitfall holds for other constructs, for example

a[t_] := Piecewise[{{{t, t}, t > 0}}, {0, 0}]

This is meant to be a vactor-valued function, but try what happens with a[t]+{1,2}.

Your solution to define f[t_?NumberQ] := dsq[c[t], q]is perfectly valid, because this way the expressions c[t]-q and c[t]+q will only be evaluated when t is a number, and there is no symbolic pre-processing. Another solution could be to restrict the definition of dst to explicit lists:

Clear[dst];
dst[p_List, q_List] := (p - q) . (p - q);
FindMinimum[dst[c[t], q], {t, .3, .301}]

Thanks for this. I think I am beginning to understand this.