I'm reading Reitz's Foundation of electromagnetic theory and in the first chapter when he presents the dot and the cross product, he gives for the equation $c= A\cdot X$ ($c$ some known scalar and $A$ some known vector) the following solution for $X$: $$X=\frac{cA}{A \cdot A}+ B$$ where $B$ can be any vector perpendicular to $A$
Then he gives for the equation $C=A×X$ (where $A$ and $C$ are known vectors) the solution: $$X=\frac{C×A}{A \cdot A}+kA$$ where $k$ can be any scalar and provided $C \cdot A=0$
I can easily verify those solutions are correct but I have noidea how to deduce them and the book does not provide the deduction's steps, so I'm asking for some clarification on the deduction of these formuli.
Thanks
I'll discuss the first example, since the reasoning in the second example is similar.
You know from the start that there will be two degrees of freedom available, and that the space of vectors perpendicular to $A$ will have two degrees of freedom. You also know that if $B$ is perpendicular to $A$ then $A\cdot B = 0$ so that if you gcan guess any one solution $X_0$, the entire set of solutions will be given by $X_0 + B$.
(Compare with the second example, where now you need to know that $A\times A= 0$ to see that you can add a multiple of $A$ to any particular solution, to get the general solution.)
So how to guess a particular solution of $C = A\cdot X$? Well, since we will be ignoring things that are perpendicular to $A$, a natural guess is $X=A$. When you try that you find $$ c = A\cdot X = A\cdot A $$ which will not be true unless you are so lucky as to have $c = A\cdot A$ in the first place.
You have to get $c$ into the picture, and the only thing you can vary is $X$, so let's try $X = cA$. This next try gives $$ c = A\cdot(cA) = c A\cdot A $$ which is not true unless $A\cdot A = 1$. But you can change $X$ once again to help with this; merely divide by $A\cdot A$. By now you have the guess $$ X =\frac{cA}{A\cdot A} $$ which now gives $c=c$ so you are home free.