Position and Displacement Vectors

In kinematics, describing the motion of an object requires knowing where the object is and how its position changes with time. These ideas are explained using two important vector quantities: the position vector and the displacement vector. Both play a fundamental role in understanding motion, velocity, and acceleration.

Position Vector

The position vector specifies the location of an object with respect to a fixed reference point, usually taken as the origin of a coordinate system. It is a vector that starts from the origin and points toward the position of the object.

If a point has coordinates (x, y) in a two-dimensional plane, its position vector () is given by

The magnitude of the position vector is given by

In three dimensions, for a point (x, y, z), the position vector is

The magnitude of the position vector represents the distance of the object from the origin, while its direction indicates where the object is located.

Displacement Vector

The displacement vector describes the change in position of an object. If an object moves from an initial position A to a final position B, the displacement vector is the vector drawn from A to B.

If the coordinates of A and B are (x₁, y₁) and (x₂, y₂), respectively, the displacement vector () is

The magnitude of the displacement vector is given by

The magnitude of displacement is the shortest distance between the initial and final positions, and its direction is always from the initial point to the final point. Displacement depends only on these two positions and not on the path taken.

Position Vector vs Displacement Vector

• The position vector represents the location of an object relative to the origin.
• The displacement vector represents the change in position between two points.
• Position vector depends on the chosen reference point, whereas displacement does not.
• Displacement can be zero even if the object has traveled a non-zero distance.

Distance vs Displacement

Distance is a scalar quantity that depends on the actual path followed by the object, while displacement is a vector quantity that depends only on the initial and final positions.

Physical Significance

The position vector helps determine where an object is at a given instant, while the displacement vector is used to define important kinematic quantities such as velocity and acceleration. These vectors form the foundation for the mathematical description of motion.

Related Articles

• Avergae Velcoity
• Motion in 2-Dimension
• Projectile motion

Solved Example

Question 1 : The position vector of a particle moving in a plane is given by . Find the displacement of the particle between t = 1 sec and t = 4 sec ?

Displacement depends only on the initial and final positions of the particle.

Given:

Position at t = 1sec :

Position at t = 4sec :

Displacement vector:

Question 2 : The position vector of a particle is given by where a and b are constants. Find the ratio of displacement to distance travelled by the particle between t=0 and t=T.

Initial position:

Final position:

Displacement magnitude:

Velocity:

Speed:

Distance travelled:

Required ratio:

Unsolved Questions

Question 1:- The position vector of a particle is given by . Find the magnitude of displacement of the particle between t = 1 sec and t = 3 sec

Question 2 :- A particle moves such that its position vector is . Find the angle made by the displacement vector with the x-axis between t=0 sec and t=2 sec.