Colliding particles

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Problem:

Two particles, $1$ and $2$, move with constant velocities $v_1$ and $v_2$. At the initial moment their radius vectors are equal to $r_1$ and $r_2$. How must these four vectors be interrelated for the particles to collide? 


Solution:

Let the particles collide at point $A$ with position vector $r_3$. 

Let $t$ is the time taken by the particles to reach $A$, and then we have $$r_3=r_1+v_1t=r_2+v_2t$$ $$\implies r_1-r_2=(v_1-v_2)t$$ $$\implies t=\frac{|r_1-r_2|}{|v_1-v_2|}.$$

From the last $2$ equations, we have $$r_1=r_2-(v_2-v_1)\frac{r_1-r_2}{|v_2-v_1|}$$, or $$\frac{r_1-r_2}{|r_1-r_2|}=\frac{v_2-v_1}{|v_2-v_1|},$$ which is the desired relationship. 


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