Two particles A and B, move with constant velocities $\overrightarrow {{v_1}} $ and $\overrightarrow {{v_2}} $. At the initial moment their position vectors are $\overrightarrow {{r_1}} $ and $\overrightarrow {{r_2}} $ respectively. The condition for particles A and B for their collision is
${\overrightarrow r _1} \times {\overrightarrow v _1} = {\overrightarrow r _2} \times {\overrightarrow v _2}$
${\overrightarrow r _1} - {\overrightarrow r _2} = {\overrightarrow v _1} - {\overrightarrow v _2}$
${{{{\overrightarrow r }_1} - {{\overrightarrow r }_2}} \over {\left| {{{\overrightarrow r }_1} - {{\overrightarrow r }_2}} \right|}} = {{{{\overrightarrow v }_2} - {{\overrightarrow v }_1}} \over {\left| {{{\overrightarrow v }_2} - {{\overrightarrow v }_1}} \right|}}$
${\overrightarrow r _1}.{\overrightarrow v _1} = {\overrightarrow r _2}.{\overrightarrow v _2}$
