How to solve $A - A \times B = C$ for vector $A$?

Question

How do I solve $A - A \times B = C$ for vector $A$? Seems so simple but I cannot get an answer.

Answer 1

Assuming $A,B,C \in \mathbb{R}^3$.
Can Solve Component Wise.
We know that $A \times B=(a_2b_3-a_3b_2)i -(a_1b_3-a_3b_1)j+(a_1b_2-a_2b_1)k$.
(See Wikipedia: Cross Product). Where i,j,k are the three dimesnional unit vectors in $\mathbb{R}^3$.

Then we can solve for A component-wise:

$a_1 - (a_2b_3-a_3b_2)=c_1$, $a_2+(a_1b_3-a_3b_1)=c_2$, and $a_3-(a_1b_2-a_2b_1)=c_3$.

Thus have three linear equations with three unkowns and can solve for A analytically.