Given is the equality
$$\vec{A}\cdot\vec{B}=c$$ What is the strategy for breaking up the scalar product and express either $\vec{A}$ or $\vec{B}$?
I know one strategy, say $\vec{A}$ is a constant vector and $\vec{B}$ is the position vector $\vec{x}$:$$\vec{A}\cdot\vec{x}=c$$ Take gradient of both sides: $$\nabla(\vec{A}\cdot\vec{x})=\nabla c,$$ then $$\vec{A}=\nabla c$$
Do you have a strategy for doing this when both $\vec{A}$ and $\vec{B}$ are not constant? How do we eliminate $\vec{A}$ form the LHS and express $\vec{B}$.
From the scalar product definition, you have that given vector A, vector B will lie anywhere in a subspace normal to A, with distance $B_A$ from the origin. $B_A$ is the projection of B onto the direction of A, and the condition is $|A| |B_A| = c$.
Say vector N is normal to vector A, then $B = (c/|A|^2)A + N$. This holds for any vector N normal to vector A; hence only the component of B into the direction of A is determined by your equation $A B = c$.