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 :

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

where

The magnitude of position vector is given by :

### Example 1: **Position and distance**

Problem : Position (in meters) of a moving particle as a function of time (in seconds) is given by :

Find the coordinates of the positions of the particle at the start of the motion and at time t = 2 s. Also, determine the linear distances of the positions of the particle from the origin of the coordinate system at these time instants.

Solution : The coordinates of the position are projection of position vector on three mutually perpendicular axes. Whereas linear distance of the position of the particle from the origin of the coordinate system is equal to the magnitude of the position vector. Now,

When t = 0 (start of the motion)

The coordinates are :

and the linear distance from the origin is :

When t = 2 s

The coordinates are :

and the linear distance from the origin is :