Shortest distance between two objects

Question

Two objects $O_1$ and $O_2$ move in parallel according to the vectors $\vec{v_1}$, and $\vec{v_2}$. Determine the shortest distance in which objects will be and the time $t$ after which it will occur.

How I can solve it?

Answer 1

Let $\vec{u_1}$ and $\vec{u_2}$ be the starting positions of the two objects. If the two objects move in parallel, $\vec{v_2} = k\vec{v_1}$. Hence the positions $\vec{x}$ of each object after time $t$ will be as follows: $$\vec{x_1}=\vec{u_1}+t\vec{v_1}$$ $$\vec{x_2}=\vec{u_2}+kt\vec{v_1}$$ The distance of the two objects is $$d=|\vec{u_1}-\vec{u_2}+t(1-k)\vec{v_1}|$$ Now you can find the minimum of $d$ to find the shortest distance between the objects.