## Quick Reference

Let **r**_{P} and **r**_{Q} be the position vectors of particles *P* and *Q* with respect to some frame of reference with origin *O*, as shown in the diagram. The position vector of *P* relative to *Q* is **r**_{P}−**r**_{Q}. If **v**_{P}=**ṙ**_{P} and **v**_{Q}=**ṙ**_{Q}, then **v**_{P} and **v**_{Q} are the velocities of *P* and *Q* relative to the frame of reference with origin *O* and **v**_{P}−**v**_{Q} is the velocity of *P* relative to *Q*. If **a**_{P}=v̇_{P} and **a**_{Q}=v̇_{Q}, then **a**_{P} and **a**_{Q} are the accelerations of *P* and *Q* relative to the frame of reference with origin *O*, and **a**_{P}−**a**_{Q} is the acceleration of *P* relative to *Q*. These quantities may be called the relative position, the relative velocity and the relative acceleration of *P* with respect to *Q*.

These notions are important when there are two or more frames of reference, each with an associated observer. For example, in a problem involving a ship and an aeroplane, the ship's captain and an observer on the land may both be viewing the aeroplane. The velocity of the aeroplane, for example, depends on which of them is making the measurement. The distinction must be made between the velocity of the aeroplane relative to the ship and the velocity of the aeroplane relative to the land.

*Subjects:*
Mathematics.