For two particles to collide, they must be at the same position at the same time $t$ ($t > 0$). Let the collision occur at time $t$. Position of particle A at time $t$: $\mathbf{r}_A = \mathbf{r}_1 + \mathbf{v}_1 t$ Position of particle B at time $t$: $\mathbf{r}_B = \mathbf{r}_2 + \mathbf{v}_2 t$ For collision, $\mathbf{r}_A = \mathbf{r}_B$: $\mathbf{r}_1 + \mathbf{v}_1 t = \mathbf{r}_2 + \mathbf{v}_2 t$ $\mathbf{r}_1 - \mathbf{r}_2 = (\mathbf{v}_2 - \mathbf{v}_1) t$ Since time $t$ is a positive scalar, the vector representing relative initial displacement $(\mathbf{r}_1 - \mathbf{r}_2)$ must be in the same direction as the relative velocity vector $(\mathbf{v}_2 - \mathbf{v}_1)$. This implies that their unit vectors must be equal: $\frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \mathbf{r}_2|} = \frac{\mathbf{v}_2 - \mathbf{v}_1}{|\mathbf{v}_2 - \mathbf{v}_1|}$,