The lines $\overline{\mathrm{r}}=\overline{\mathrm{a}}+\mathrm{t}\overline{\mathrm{b}}, \overline{\mathrm{r}}=\overline{\mathrm{c}}+{\mathrm{t}}^1\overline{\mathrm{d}}$ are coplanar if

$$\begin{gathered} \overline{\mathrm{r}}=\overline{\mathrm{a}}+\mathrm{t}\overline{\mathrm{b}}, \overline{\mathrm{r}}=\overline{\mathrm{c}}+{\mathrm{t}}^1\overline{\mathrm{d}} \\ \text{ We know that }\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{c}}\text{ are the fixed points lying on the line.} \\ \text{ And }\overline{\mathrm{b}},\overline{\mathrm{d}}\text{ are the direction ratios lying of the line.} \\ \text{ For coplanarity we need third vector lying on the plane which can be } \\ \overrightarrow{\mathrm{m}}=\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{c}} \\ \text{ Hence for Coplanar Lines } \\ \Rightarrow \overrightarrow{\mathrm{m}}\cdot (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{d}})=0 \end{gathered}$$

$\left(\overline{\mathrm{a}}-\overline{\mathrm{c}}\right)·\left(\overline{\mathrm{b}}\times \overline{\mathrm{d}}\right)=0$