What is the locus such that any vector from it has a given dot product with the given vector?

Question

Consider a given vector $a$ and scalar $d$. What is the set $X$ such that for any $x \in X$ their dot product equals $d$ : $\forall x \in X: x \cdot a = d$ ?

Answer 1

It's easy to write down one point $p = \frac{a * d}{|a|^2}$ in this set. For any other point $q$, we have $(p - q) \cdot a = 0$, so the set of vectors $p - q$ is precisely the set of vectors orthogonal to $a$.