What can we say if the inner products of two vectors ($u$ and $v$) with another vector ($x$) are equal? ($x$ is an eigenvector)

Question

We know that $x$ is an eigenvector and the inner products of $x$ with $u$ and $v$ are equal, I mean:

$\langle u,x\rangle \;= \; \langle v,x \rangle$

On the other hand, we know that $v = [1,1,...,1]$.

$u$ and $v$ and $x$ are defined on $\mathbb{R}^d$. And by the inner-product, I mean the sum of elements of the output of the element-wise multiplication. What can we say about the relation between $u$ and $v$?

Answer 1

$\langle w, x \rangle$ is a scalar multiple of the length projection of $w$ onto $x$. So your two inner products are equal if and only if $u$ and $v$ have the same projection onto $x$, or equivalently, $u-v$ is orthogonal to $x$ (which can be easily seen by $\langle u-v, x\rangle = \langle u, x \rangle - \langle v, x \rangle = 0$).