I read in a book that the cross product $cross(a,b) = 0$ contains only two independent equations. I thought this cross product gave us three equations, but are they not all independent? How do I get the two independent equations from the cross product above?
Clarification: the vectors $a$ and $b$ are supposed to be perpendicular, so the constraint equation $cross(a,b) = 0$ is supposed to give two constraint equations.
If you have an equation $\omega\times x = b$, you can rewrite it as a multiplication by a matrix $B$ (its coefficients are defined in terms of coefficients of $\omega$). This transformation is either given in your textbook or you can try to deduce this matrix yourself.
When you study this $3\times 3$ matrix, you can see that its rank is only $2$ (as long as $\omega\ne0$). Therefore, the identity $Bx=b$ is not three, but only two independent linear equations.