- #1

mahler1

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Let ##C## be a curve that doesn't pass through the origin and let ##P## be the closest point on the curve to the origin. Prove that the tangent to ##C## at ##P## is orthogonal to the vector ##P##.

The attempt at a solution.

Suppose ##P=\gamma(t_0)##, I want to show that ##<\gamma'(t_0),\gamma(t_0)>=0##. I am pretty lost with the exercise and I don't know why they mention the curve doesn't pass through the origin or the fact that ##P## is the closest point to the origin, are those hypothesis necessary?. I would appreciate some suggestions and maybe an intuitive idea of why these two vectors are orthogonal. By the way, in my class we are always working with curves parametrized by the arc lenght, so maybe I have to use the fact that ##|\gamma'|=1##.