System of two equations with 3 unknowns and parameters Is there a way to solve this for $c_1$, $c_2$, $c_3$ in terms of $a$'s and $b$'s?
$$
\begin{cases}
a_1c_1+a_2c_2+a_3c_3=0 \\ 
b_1c_1+b_2c_2+b_3c_3=0
\end{cases}
$$

The system tells you that the vector $\mathbf c$ shall be normal to the vectors $\mathbf a$ and $\mathbf b$. Therefore put $\mathbf c= \lambda \mathbf a \times \mathbf b$.
