In a vector space, given 3 vectors $\displaystyle a,b,c$ I want to find all solutions of $\displaystyle \left\langle x,a \right\rangle c = b$.

So far this is my approach, I am just a little hestiant to say I have found all solutions:

If we assume $\displaystyle a, b,$ and $\displaystyle c \neq 0$. Then $\displaystyle \left\langle x,a \right\rangle c = b \implies |\left\langle c,b \right\rangle| = ||c|| \cdot ||b||$ because we know equality of Cauchy- Schwarz holds iff one vector is a scalar multiple of the other.

So

$\displaystyle \frac{|\left\langle c,b \right\rangle|}{||c||^2} = |\left\langle x,a \right\rangle|$

Using this fact I know $\displaystyle x = \frac{|\left\langle c,b \right\rangle| b}{\left\langle b,a \right\rangle \left\langle c,c \right\rangle}$ is a solution.

The cases when a,b or c is 0 are less interesting I think. What do you think?