I'm working on a question in a Mechanics textbook that reads:
Assume that two vectors A and B are known. Let $C$ be an unknown vector such that $\vec{A} \cdot \vec{C} = u$ is a known quantity and $\vec{A} \times \vec{C} = \vec{B}$. Find $\vec{C}$ in terms of $\vec{A}$, $\vec{B}$, $\vec{u}$, and the magnitude of $\vec{A}$.
I had trouble figuring this one out and looked at the solution, and after puzzling it over for a while most everything makes sense to me except the terms that are added to the $C_x$ and $C_y$ terms. I see that they reduce to unit vectors so I can tell that their purpose is to correct the orientation of the components, because they don't depend on the $x$ and $y$ axes but on vectors given in the problem. My question is, why exactly those terms? Why does the $C_x$ component go in the $\vec{A}$ direction, and the $C_y$ component go in the $\vec{B} \times \vec{A}$ direction? How can this be determined from the problem?