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Applications of Horizontal and Vertical Components

Two particles...

Question

Two particles, $1$ and $2$ moves 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.

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Solution

..Velocity of B w.r.t A
$_{B . A} =_{2} -_{1}$
Position of B w.r.t A.
$_{B A} =_{B} -_{A}$
particle B collide with A if the velocity of B w.r.t A is directed towards A.
$\frac{_{2} -_{1}}{|_{2} -_{1} |} = \frac{_{2} -_{1}}{|_{2} -_{1} |}$

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