If a body is undergoing both translation and rotation, then the vectors that describe the velocity of each point can be different. Figure 2↓ shows an illustration of this scenario. A and B are two arbitrary points on the rigid body. A frame of reference can be attached to this rigid body. Frame a can be defined with a₁, a₂ and a₃ as unit vectors along each axis and point A as the origin. The angular velocity of frame a with respect to inertial or fixed frame (xyz) is denoted by ω_a ⁄ I. One can write the following relation between position vectors and point A and B.

Since velocity is the time derivative of position vector, this equation becomes where (d)/(dt)P_OB = V_B and (d)/(dt)P_OA = V_A since they are with respect to the frame where the origin O is located (notice O is in the subscript). (d)/(dt)P_AB is the relative velocity of B with respect to A, the nomenclature V_B ⁄ A to show this term; therefore Equation (5↑) is true for any arbitrary point A and B, regardless of if they are on the same rigid body or not. If A and B are on the same rigid body then the distance between them is constant. Let e be a unit vector in the direction of P_AB, so that the distance can be defined as where |AB| is the magnitude of P_AB. Since A and B are on the same rigid body so that|AB| is constant, then Using the chain rule, this becomes Since |AB| is constant, the underlined term in equation (8↑) is zero. Also the derivative of a unit vector is Substituting equation (9↑) into (8↑) and Since |AB|e is P_AB, then