Displacement and Vectors- Definition, Explanation

Introduction: 

In mechanics(motion, force, etc.), we come across various measurable, derivable and formulating quantities. To measure or calculate a particular set of quantities, you have to understand which category they belong to. In this session, we will discuss and understand those quantities by which we can calculate the motion. 

Mainly we consider a quantity as 

1. Scalars  
2. Vectors 

Explanation: 

Scalar Quantity: 

Scalars are the physical quantities that have only magnitude but not direction. 

Scalars have only value but not direction. 

For example, a car is moving at a speed of 35 m/s. 

Here only the value of the speed is mentioned as 35 m/s, but it is not said in which direction the car is moving. 

Examples of Scalar Quantity: 

• Speed 
• Distance 

Vector Quantity: 

A vector that has both directions as well as magnitude. 

Example: A car is moving with a velocity of 35 m/s in the east direction.  

Here unlike the scalar, the value and the direction are mentioned.  

If we consider coordinates such as the x-axis and Y-axis, usually the direction is mentioned as positive or negative X-axis as well as positive or negative Y-axis. 

Examples of Vector Quantity: 

• Linear momentum 
• Acceleration 
• Displacement 

Position Vectors 

Consider using the coordinates say plane (2-dimension) or space (3-dimension). 

Say there is a vector P present in the space, how can we locate the vector? 

Using the position vector or representing the vectors in the Space by using some coordinates, we can say the position of the vector. 

Let us consider i, j, k as the vectors coordinates for the axis (x, y, z). As we are taking the position vector in space, it has three dimensions (x, y, z). 

Therefore, 

The position vector

usually vector are denoted with “^” then, 

Where, 

= unit vector along x-direction. 

= unit vector along the y-direction. 

= unit vector along z-direction. 

Where a, b, c are coordinates for x, y, z. 

Using this equation, we can find the position vector in the given instance of time. 

Displacement Vector: 

In the formula mentioned earlier, we considered that the position vector is not moving, and it is at one point at the instance of time. What if the vector is moving from one point to another in the case of time? 

We have to consider the displacement vector if the vector moves from one point to another in time.  

Suppose a vector P is in position A in time t₁ and moved to the position B in time t₂. To find the position of the vector at different points at different instances of time,  

we have, 

at position A the vector

Therefore, position vector

is given by 

Displacement: 

Let us consider a person moving from his home to the park. 

Assume his home is at point A and playground is at point B. 

Displacement is defined as the path or length from the initial point to the final point. The displacement units are meters(m) in the SI system and centi-meter(cm) in the CGS system. 

Note: Displacement is a vector quantity; it also should have direction along with the magnitude. 

Example: 

A car is moving from point A to point B, the displacement is 15 m in positive x-direction and moving from point B to point C 10 min negative x-direction. A B and C are colinear (which are in the same line). 

What is the total displacement? 

Displacement from A to B is 15 m in positive x-direction so consider it as +15 m. 

Displacement from B to C is 10 m in negative x-direction so consider it as -10 m. 

Total displacement = +15 m+(–10 m) 

Displacement = +5 m towards positive × axis 

Displacement = Final Position-Initial 

Position=Change in position 

We can say 

Time and Time Interval: 

Time is considered between the events. Time plays a very vital role in physics, especially in mechanics. 

Unit of time is seconds(s) in both SI and CGS systems. 

In physics, time appears as” rate”, like the rate of change. 

Time interval is the phenomenon where we consider the events happening and changing with time.  

Example: The time interval for traveling from one place to another place.  

We can express time intervals for any occasion and any outcome. 

Let us say an event started at the time t₁ and completed at time t₂. 

So, the change in time is given by t = t₂ –  t₁ 

Summary:

• Scalars: Scalar quantity is defined as the physical quantity with magnitude and no direction.
• Vectors: A vector quantity is defined as the physical quantity that has both direction as well as
  magnitude.
• Position Vectors: The position vector is used to specify the position of a certain body.
  The position vector

usually vector is denoted with “^” then.

• Displacement Vector: The change in the position vector of an object is known as the displacement vector

Therefore position vector

• Displacement: Displacement is defined as the path or length from the initial point to the final point. Units of the displacement are meters(m) in the SI system and centi-meter(cm) in the CGS system.

Note: Displacement is a vector quantity; it also should have direction along with the magnitude.

• Time and Time Intervals: Time is considered between the events, time plays very vital role in
  physics, especially in mechanics.
• Unit of time is seconds(s) in both SI and CGS systems.
• Time interval is the phenomenon where we consider the events happening and changing with
  time.