This problem is from Calculus of Several variables by Serge Lang. It says
Let $\vec{X}(t)$ be a differentiable curve (by curve he probably means a vector valued function) defined on an open interval, Let $Q$ be a point that is not on the curve.
The first question says to find the formula for the distance between $Q$ And a point on the curve.
this one is pretty straightforward, the formula is $$D(Q,X(t))=\sqrt{(\vec{Q}-\vec{X}(t))\cdot (\vec{Q}-\vec{X}(t))}$$ The second question: If $X(t_o)$ is a point on the curve such that the distance formula tends to its minimum,Prove that $\vec{Q}-\vec{X}(t_o)$ is perpendicular to the curve at the point $\vec{X}(t_o)$
So this one i don’t have any clue to attack it, i just know that proving the statement in the second question is equivalent to proving that $$(\vec{Q}-\vec{X}(t_o)) \cdot\vec{X’}(t_o)=0 $$