The displacement time equations of two particles are: $$s_1 = 2t-4t^2$$ $$s_2 = -2t + 4t^2$$
By differentiating we can find $v_1,v_2$. Subtracting them gives: $v_1-v_2 = 4-16t$ Clearly, this continuously decreases with time, so the relative velocity decreases. But if we subtract $v_2 - v_1$, this increases with time ($16t-4$). It seems so simple and thus silly, but yet it's confusing me. Any help is appreciated!
Velocity is a vector. It has a direction.
You don't really see the direction here because it is a $1$D problem. But the sign indicates whether the direction is toward negative numbers (left) or positive (right).
Relative velocity is the velocity you would see if you sat on one particle and watched the other. $v_1 - v_2$ is the velocity of particle $1$ if you sat on particle $2$. $v_2 - v_1$ is the velocity of particle $2$ if you sat on particle $1$. These have the same magnitude, but opposite direction.
The magnitude indicates relative speed. When the magnitude is $0$, they are not moving with respect to each other. When the magnitude is decreasing, they are approaching the same speed, or slowing with respect to each other. When the magnitude increases, the relative speed is increasing.
