Find all possible vectors that satisfy an inner product

Question

Suppose I have three vectors ($\mathbf{v}$, $\mathbf{w}$, $\mathbf{x}$) and a scalar $\alpha$ such that
$$
(\mathbf{x} \circ \mathbf{v})^\top \mathbf{w} = \alpha
$$
which is the inner product between $(\mathbf{x} \circ \mathbf{v})$ and $\mathbf{w}$, with $\circ$ denoting element-wise multiplication. The vector $\mathbf{w}$ is known and constrained such that $\sum w_i = 1$, the vector $\mathbf{x}$ is known, and the scalar $\alpha$ is known. I need to find all possible solutions for $\mathbf{v}$ such that the inner product holds. How do I go about doing this? Also, once I find that space how do I go about randomly sampling from it?

Answer 1

Since $(x\circ v)^\top w=\sum_ix_iv_iw_i=v^\top(x\circ w)$, the solution set is given by $\dfrac{\alpha x\circ w}{\|x\circ w\|^2}+\{x\circ w\}^\perp$. In other words, if $u$ is any random vector that is orthogonal to $x\circ w$, then $v=\dfrac{\alpha x\circ w}{\|x\circ w\|^2}+u$ is a solution to $(x\circ v)^\top w=\alpha$.