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Submanifold, a vector is orthogonal to the tangent space


New member
Jan 11, 2014
Could you help me solve this problem?

Let $M$ be a submanifold of dimension $d ={1,...,n-1}$ class $\mathcal{C}^1$ in $\mathbb{R}^n$.

Fix $a \in \mathbb{R}^n \setminus M$.

Let $x \in M$ be such that $||x-a||=\inf \{||y-a|| \ : \ y \in M \}$, where $|| \cdot ||$ is a Euclidean norm in $\mathbb{R}^n$.

Prove that $x-a$ is orthogonal to $T_xM$.

$T_xM$ is the tangent space.

Thank you for your help.


Staff member
Feb 24, 2012
Hello and welcome to MHB, Jakob! (Smile)

Can you post what you have tried thus far so our helpers know where you are stuck and can best assist you?

Otherwise you may get suggestions on how to begin that you have already tried with no success, and this would waste your time and that of our helpers.