Why distance between 2 particles is independent of frame?

Question

I was solving question in kinematics related to minimum distance between particles there they said that distance between two particles is independent of frame the distance they both measure between them is same for both(velocity<<c) of them i know this is silly question Actually i was asking here in the question( image) if we find the minimum distance between them it is found to be same from both the frame

Answer 1

Consider two particles $A$ and $B$ in frame $S$. The positions of each particle are given by $\vec{r}_A$ and $\vec{r}_B$ respectively. The distance $d$ between the two particles is given by $$d=|\vec{r}_A-\vec{r_B}|.$$

Now consider particles $A$ and $B $ in frame $S'$ whose origin is displaced from that of frame $S$ by $\vec{r}$. The positions of $A$ and $B$ in frame $S'$ are now given by $$\vec{r}_A'=\vec{r}_A+\vec{r},$$ $$\vec{r}_B'=\vec{r}_B+\vec{r}.$$ The distance $d'$ between the two particles in frame $S'$ is now given by $$d'=|\vec{r}_A'-\vec{r}_B'| = |(\vec{r}_A+\vec{r}) -(\vec{r}_B+\vec{r})|=|\vec{r}_A-\vec{r_B}|.$$

Hence we see that $d=d'$, thus proving that the distance between 2 particles is independent of the frame of reference.

Note: I have assumed that the frames are stationary relative to each other.

Answer 2

In classical mechanics length and time intervals are the same for all frames of reference. It follows that the minimum distance you calculated will be the same in all frames of reference.