The direction cosines of two lines are related by $\mathrm{l}+\mathrm{m}+\mathrm{n}=0$ and ${\mathrm{al}}^2+{\mathrm{bm}}^2+{\mathrm{cn}}^2=0$. If the lines are parallel and $\mathrm{abc}\ne 0$ then

For n=$-(\mathrm{l}+\mathrm{m})$ , the second relation gives
 ${\mathrm{al}}^2+{\mathrm{bm}}^2+\mathrm{c}{(\mathrm{l}+\mathrm{m})}^2=0$
  $\mathrm{or} (\mathrm{a}+\mathrm{c}){\mathrm{l}}^2+2\mathrm{clm}+(\mathrm{b}+\mathrm{c}){\mathrm{m}}^2=0$
For parallel lines, the two roots must be equal
$$\begin{gathered} \Rightarrow 4{\mathrm{c}}^2-4(\mathrm{b}+\mathrm{c})(\mathrm{a}+\mathrm{c})=0 \\ \Rightarrow \mathrm{ab}+\mathrm{bc}+\mathrm{ca}=0 \end{gathered}$$