# Position, Displacement and Velocity Vectors

## Position Vector

A vector that extends from a reference point to the point at which particle is located is called position vector.

Let r be the position vector of a particle P located in a plane with reference to the origin O in x-y plane.

## Displacement Vector

Displacement vector is that vector which tells how much and in which direction an object has changed its position in a given time interval.

Let an object moving in the x-y plane. Let it is at point P at any instant t₁ and at point Q at any later time t₂.

In ∆OPQ,

OP + PQ = OQ

PQ = OQ – OP

∆r = r₂ – r₁

If the position coordinate of point P and Q are (x₁, y₁) and (x₂, y₂) respectively, then

r₁ = x₁ i^ + y₁ j^

r₂ = x₂ i^ + y₂ j^

Displacement Vector ∆r = r₂ – r₁

∆r = (x₂ – x₁)i^ + (y₂ – y₁) j^

For Three Dimensions

∆r = (x₂ – x₁)i^ + (y₂ – y₁) j^ + (z₂ – z₁)k^

## Velocity Vector

The rate of change of displacement of an object in a particular direction is called its velocity.

It is of two types.

### (I) Average Velocity

It is defined as the ratio of the displacement and the corresponding time interval.

### (II) Instantaneous Velocity

The velocity at an instant of time t is known as instantaneous velocity.

The average velocity will become instantaneous, if ∆t approaches to zero.

The instantaneous velocity is expressed as-