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Two particles A and B, move with constant velocities $\vec u_1$ and $\vec u_2$. At the initial moment their position vectors are $\vec r_1$ and $\vec r_2$ respectively. The condition for particles A and B for their collision is:
A
$\vec r_1 - \vec r_2 = \vec u_1 - \vec u_2$
B
$\frac{\vec r_1 - \vec r_2}{|\vec r_1 - \vec r_2|} = \frac{\vec u_2 - \vec u_1}{|\vec u_2 - \vec u_1|}$
C
$\vec r_1\cdot\vec u_1 = \vec r_2\cdot\vec u_2$
D
$\vec r_1\times\vec u_1 = \vec r_2\times\vec u_2$
Detailed Solution
For collision, the velocity of B relative to A must point along the position of A relative to B.
Direction of relative position of A w.r.t. B: $\frac{\vec r_1 - \vec r_2}{|\vec r_1 - \vec r_2|}$
Direction of velocity of B w.r.t. A: $\frac{\vec u_2 - \vec u_1}{|\vec u_2 - \vec u_1|}$
Condition: $\frac{\vec r_1 - \vec r_2}{|\vec r_1 - \vec r_2|} = \frac{\vec u_2 - \vec u_1}{|\vec u_2 - \vec u_1|}$
Direction of relative position of A w.r.t. B: $\frac{\vec r_1 - \vec r_2}{|\vec r_1 - \vec r_2|}$
Direction of velocity of B w.r.t. A: $\frac{\vec u_2 - \vec u_1}{|\vec u_2 - \vec u_1|}$
Condition: $\frac{\vec r_1 - \vec r_2}{|\vec r_1 - \vec r_2|} = \frac{\vec u_2 - \vec u_1}{|\vec u_2 - \vec u_1|}$