Kinematics of a point

The position of a point relative to the origin is represented by the free vector r. This is rep­

resented by the product of the scalar magnitude, r, and a unit vector e

The change in a unit vector is due only to a change in direction since by definition its magnitude is a constant unity. From Fig. A it is seen that the magnitude of de is 1.dθ and is in a direction normal to e. The angle dθ can be represented by a vector normal to both e and de.

It is important to know that finite angles are not vector quantities since they do not obey the parallelogram law of vector addition. This is easily demonstrated by rotating a box about orthogonal axes and then altering the order of rotation.

Non-vectoral addition takes place if the axes are fixed or if the axes are attached to the box. 

The change in the unit vector can be expressed by a vector product thus

Fig A.

where   

is the angular velocity of the unit vector e. Thus, we may write

…………………(A)

where the partial differentiation is the rate of change of r as seen from the moving axes. The

form of above equation is applicable to any vector V expressed in terms of moving coordinates, so

…………………(B)

Acceleration is by definition

And by using equation (B)

Using Equation (A)