Can I express $\vec{v}$ in $\vec{v} \cdot \vec{x} = c$ as $\vec{v}= c \vec{x}/ \left| \vec{x} \right|^{2}$?

Question

Suppose I have a formula like this

$ \vec{v} \cdot \vec{x} = c $

where $\vec{v}$ and $\vec{x}$ are non-zero vectors (dimension greater than $1$) and $c$ is a non-zero scalar. Can I express $\vec{v}$ as

$\vec{v} = c \vec{x} / \left| \vec{x} \right|^{2}$

If I cannot, is there any method to express $\vec{v}$ in term of $\vec{x}$ ? Thank you very much for your help and I am waiting for your reply.

Answer 1

If $\vec x$ is a fixed (nonzero) vector, the set of vectors $\vec v$ satisfying your equation is a line (in the plane) or plane (in 3-space) with normal vector $\vec x$ passing through the point $c\vec x/\|\vec x\|^2$.

Answer 2

Let $\vec y$ be any vector that is perpendicular to $\vec x$. That is, $\vec x\cdot\vec y=0$.

Then, the vector $\vec z=\frac{c\vec x}{|\vec x|^2}+\vec y$ satisfies the expression $\vec z\cdot \vec x=c$.

Hence, given $\vec v\cdot\vec x=c$, we cannot deduce that $\vec v=\frac{c\vec x}{|\vec x|^2}$.