A carrom board ( 4 ft × 4 ft square) features the queen at its...

Let the carrom board be represented in a coordinate system with its corners at $(0,0)$, $(4,0)$, $(4,4)$, and $(0,4)$.

The initial position of the queen is at the centre of the board, which has coordinates $(2,2)$.

The final position of the queen is the hole behind the striking line. Without loss of generality, let this hole be at the origin $(0,0)$.

Although the queen travels to the front edge and rebounds, displacement is defined as the shortest straight-line distance between the initial and final positions, independent of the actual path taken.

Displacement vector $\vec{s} = \vec{r}_f - \vec{r}_i = (0 - 2)\hat{i} + (0 - 2)\hat{j} = -2\hat{i} - 2\hat{j}$

Magnitude of displacement $|\vec{s}| = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \text{ ft}$.