1.11 Position vector

Position vector is a convenient mathematical construct to encapsulate the twin ideas of magnitude (how far?) and direction (in which direction?) of the position, occupied by an object.

Position vector

Position vector is a vector that extends from the reference point to the position of the particle.

Generally, we take origin of the coordinate system as the reference point.

It is easy to realize that vector representation of position is appropriate, where directional properties of the motion are investigated. As a matter of fact, three important directional attributes of motion, namely displacement, velocity and acceleration are defined in terms of position vectors.

Consider the definitions : the “displacement” is equal to the change in position vector; the “velocity” is equal to the rate of change of position vector with respect to time; and “acceleration” is equal to the rate of change of velocity with respect to time, which, in turn, is the rate of change of position vector. Thus, all directional attributes of motion is based on the processing of position vectors.

Position vector in component form

One of the important characteristics of position vector is that it is rooted to the origin of the coordinate system. We shall find that most other vectors associated with physical quantities, having directional properties, are floating vectors and not rooted to a point of the coordinate system like position vector.

Recall that scalar components are graphically obtained by dropping two perpendiculars from the ends of the vector to the axes. In the case of position vector, one of the end is the origin itself. As position vector is rooted to the origin, the scalar components of position vectors in three mutually perpendicular directions of the coordinate system are equal to the coordinates themselves. The scalar components of position vector, r , by definition in the designated directions of the rectangular axes are :

$$\begin{array}{l}x=r\cos \alpha \\ y=r\cos \beta \\ z=r\cos \gamma \end{array}$$

and position vector in terms of components (coordinates) is :

$$\begin{array}{l}\mathbf{r}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\end{array}$$

where $\mathbf{i}$ , $\mathbf{j}$ and $\mathbf{k}$ are unit vectors in x, y and z directions.

The magnitude of position vector is given by :

$$\begin{array}{l}r=\left|\mathbf{r}\right|=\surd (x^2+y^2+z^2)\end{array}$$

Motion types and position vector

Position is a three dimensional concept, requiring three coordinate values to specify it. Motion of a particle, however, can take place in one (linear) and two (planar) dimensions as well.

In two dimensional motion, two of the three coordinates change with time. The remaining third coordinate is constant. By appropriately choosing the coordinate system, we can eliminate the need of specifying the third coordinate.

In one dimensional motion, only one of the three coordinates is changing with time. Other two coordinates are constant through out the motion. As such, it would be suffice to describe positions of the particle with the values of changing coordinate and neglecting the remaining coordinates.

A motion along x –axis or parallel to x – axis is, thus, described by x - component of the position vector i.e. x – coordinate of the position as shown in the figure. It is only the x-coordinate that changes with time; other two coordinates remain same.

The description of one dimensional motion is further simplified by shifting axis to the path of motion as shown below. In this case, other coordinates are individually equal to zero.

$$\text{x = x; y = 0; z = 0}$$

In this case, position vector itself is along x – axis and, therefore, its magnitude is equal to x – coordinate.

Examples