The condition that the lines $\overline{\mathrm{r}}=\overline{\mathrm{a}}+\mathrm{t}\overline{\mathrm{b}} , \overline{\mathrm{r}}=\overline{\mathrm{c}}+\mathrm{s}\overline{\mathrm{d}}$ to intersect each other is

$$\begin{gathered} \overrightarrow{\gamma }=\overrightarrow{a}+t\overrightarrow{b},\overrightarrow{r}=\overrightarrow{c}+s\overrightarrow{d} \\ \text{ these two lines are inter sect each if the given lines are co-planer. } \\ \text{ and the lines are co-planar if }\overrightarrow{a}-\overrightarrow{c},\overrightarrow{b},\overrightarrow{d}\text{ are co-planer. } \\ \Rightarrow (\overrightarrow{a}-\overrightarrow{c})\overrightarrow{b}\times \overrightarrow{d}=0 \\ \Rightarrow \text{ The required condition is }[\overrightarrow{c}-\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}]=0 \end{gathered}$$