I'm having a hard time with a analytic geometry problem. It says: Given the vectors $\vec u = (0,1,-2)$, $\vec v = (3,-6,-3)$ and $\vec w = (0,-1,3)$, determine the vector $\vec a$, orthogonal to $\vec w$, such that $\vec a \times \vec u = \vec v$. I already found the outer product of $\vec a \times \vec u$ using determinants, but I'm not sure how to go on and figure out the variables such that $\vec a \bullet \vec w = 0$.
Thanks in advance.
I would start with the fact that $a \cdot w =0$, this says that $a$ is of the form $(x,3z,z)$. Then take a cross product of a vector of this form with $u$, solving for $x$ and$ z$, to make the result $v$.