If the translation of a vector remains orthogonal to a translation of a space?

Question

Let $V$ be a vector space and $W$ a subset of it which is not necessarily a vector space. $v$ is orthogonal to $W$.

Given some vector $u$ one can guarantee that $v+u$ is still orthogonal to $W+u$?

Observe that $W+u=\{w+u| \ w\in W\}.$

By drawing the figure it seems yes, however I could not prove that. Any comment or answer?

Answer 1

This is not true.

Consider a simple 2D example: take $v = (1,0)$, $W = \{(0,1)\}$ and $u = (1,1)$. Then $v \perp W$, but $v+u=(2,1) \not\perp \{(1,2)\}=W+u$

Answer 2

Let $\langle v,w\rangle = 0$ for all $w\in W$. For each $u$, $\langle u+v,u+w\rangle = \langle u,u\rangle +\langle u,w\rangle + \langle v,u\rangle + \langle v,w\rangle$, since the scalar product is bilinear.

Answer 3

What if $v=0?$. Then $v\perp W$ but $u$ not necessarily perpendicular to $u+W$