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Question:

Two particles A and B move with constant velocities ⃗v₁ and ⃗v₂. At the initial moment their position vectors are ⃗r₁ and ⃗r₂ respectively. The condition for particles A and B for their collision is:

⃗r₁⋅⃗v₁=⃗r₂⋅⃗v₂

⃗r₁−⃗r₂=⃗v₁−⃗v₂

‖⃗r₁−⃗r₂‖=‖⃗v₂−⃗v₁‖

⃗r₁×⃗v₁=⃗r₂×⃗v₂

Solution:

For the collision the final position of the both particles should be equal.hence,
⃗r₁+⃗v₁t=⃗r₂+⃗v₂t
or
⃗r₁−⃗r₂=(⃗v₂−⃗v₁)t
or
⃗r₁−⃗r₂=(⃗v₂−⃗v₁)∣∣⃗r₁−⃗r₂∣∣∣∣⃗v₂−⃗v₁∣∣
or
∣∣⃗r₁−⃗r₂∣∣∣∣⃗r₁−⃗r₂∣∣=∣∣⃗v₂−⃗v₁∣∣∣∣⃗v₂−⃗v₁∣∣