Q.Define the position vector and the displacement vector of a particle. If a particle's position vectors at two instants are $\vec r_1$ and $\vec r_2$, derive an expression for its displacement in terms of $\vec r_1$ and $\vec r_2$, and explain why displacement does not depend on the choice of origin even though the position vector does.

[!TLDR] Displacement is the vector from the earlier position to the later position. [!ANSWER] $\Delta\vec r = \vec r_2 - \vec r_1$; it is origin-independent because any shift of origin adds the same constant to both $\vec r_1$ and $\vec r_2$, which cancels in the subtraction.

The position vector $\vec r$ of a particle at any instant is the vector drawn from a chosen fixed origin $O$ to the particle's location at that instant. The displacement of the particle between an earlier instant $t_1$ (position vector $\vec r_1$) and a later instant $t_2$ (position vector $\vec r_2$) is the vector drawn directly from the earlier location to the later location, and is obtained by vector subtraction: $\Delta\vec r = \vec r_2 - \vec r_1$. If the origin is now shifted to a new point $O'$, every position vector changes by the same constant vector $\vec c$ (the vector from $O'$ to $O$): the new position vectors become $\vec r_1' = \vec r_1 + \vec c$ and $\vec r_2' = \vec r_2 + \vec c$. The new displacement is $\vec r_2' - \vec r_1' = (\vec r_2+\vec c)-(\vec r_1+\vec c) = \vec r_2 - \vec r_1$ — identical to before, because the constant $\vec c$ cancels exactly in the subtraction. So while the position vector of a single point depends on where the origin is placed, the displacement between two points does not. [!ANSWER] $\Delta\vec r = \vec r_2 - \vec r_1$; it is origin-independent because any shift of origin adds the same constant to both $\vec r_1$ and $\vec r_2$, which cancels in the subtraction.

Define both vectors, write the subtraction, then show algebraically that an origin shift adds an identical constant to both terms and cancels.

• Confusing displacement (a vector, straight-line) with distance (a scalar, actual path length).\n- Believing displacement changes if the origin is moved.