Two particles moving at consant velocity: how close do they get?

Question

A particle was at point $P_1$ at time $t_1$ and is moving at the constant velocity $\vec{v}_1$. Another particle was at $P_2$ at $t_2$ and is moving at the constant velocity $\vec{v}_2$. How close did the particles get to each other and at what time? What conditions are needed for a collision?

Answer 1

There's collision at time t if : $\vec{OP_1}+(t-t_1)\cdot \vec{V_1}=\vec{OP_2}+(t-t_2)\cdot \vec{V_2}$, where O is the origin.

If not, $\vec{OP_1}+(t-t_1)\cdot \vec{V_1}-(\vec{OP_2}+(t-t_2)\cdot \vec{V_2})=\alpha\cdot\vec{V_1}\wedge\vec{V_2}+\beta(\vec{V_1}+\vec{V_2}) $

Therefore you have a system with 3 equations (projection on $\vec{x},\vec{y},\vec{z}$) and 3 unknowns $(t,\alpha, \beta)$

EDIT
By the way searching for numerical solutions, I found this topic which is exactly the
same problem : here