a
\(\begin{array}{l}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Let\,the\,particles\,A\,and\,B\,collide\\
at\,time\,t.\,For\,their\,collision,\,the\,position\\
vectors\,of\,both\,particles\,sholud\,be\,\\
same\,at\,time\,t,\,i.e.,\\
{{\vec r}_1} + {{\vec v}_1}t = {{\vec r}_2} + {{\vec v}_2}t\,;\,{{\vec r}_1} - {{\vec r}_2} = {{\vec v}_2}t - {{\vec v}_1}t\\
 = \left( {{{\vec v}_2} - {{\vec v}_1}} \right)t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i \right)
\end{array}\)

\(\begin{array}{l}
Also,\,\left| {\,{{\vec r}_1} - {{\vec r}_2}} \right| = \left| {{{\vec v}_2} - {{\vec v}_1}} \right|t\,\,or\,t = \frac{{\left| {\,{{\vec r}_1} - \vec r} \right|}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}\\
Substituting\,this\,value\,of\,t\,in\,eqn.\,\left( i \right),\\
we\,get\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\vec r}_1} - {{\vec r}_2} = \left( {{{\vec v}_2} - {{\vec v}_1}} \right)\frac{{\left| {\,{{\vec r}_1} - \vec r} \right|}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}\\
or\,\frac{{\left| {\,{{\vec r}_1} - {{\vec r}_2}} \right|}}{{\left| {\,{{\vec r}_1} - {{\vec r}_2}} \right|}} = \frac{{\left( {{{\vec v}_2} - {{\vec v}_1}} \right)}}{{\left| {{{\vec v}_2} - {{\vec v}_1}} \right|}}
\end{array}\)