Relative displacement refers to the change in position of one object in relation to another object. It is a vector quantity, meaning it has both magnitude and direction.

To calculate relative displacement, you subtract the initial position vector of the second object from the final position vector of the first object. This can be mathematically represented as:

where $\mathbf{r}_{A,f}$ is the final position vector of object A and $\mathbf{r}_{B,i}$ is the initial position vector of object B.

For instance, consider the following example: If object A moves from position $(2,3)$ to position $(5,7)$ and object B moves from position $(1,1)$ to position $(4,5)$, the relative displacement of object A with respect to object B can be calculated as follows:

This result indicates that object A has moved $4$ units to the right and $6$ units upwards relative to object B.

It is important to note that relative displacement can be negative, which occurs when the two objects move in opposite directions. For example, if object A moves from position $(2,3)$ to position $(5,7)$ and object B moves from position $(6,5)$ to position $(3,1)$, the relative displacement of object A with respect to object B is:

In this case, the negative value indicates that object A has moved $1$ unit to the left and $2$ units upwards in relation to object B.

Understanding relative displacement is crucial in physics and engineering, as it allows us to describe the motion of objects in relation to one another effectively.