Position Vector and Displacement Vector - JEE

VSAT 2022

What is Position and Displacement Vector?

In general, a position vector is utilised to find the location of an object relative to another object. This generally begins at the origin and then ends at the other arbitrary points. That’s why these vectors are utilised to determine the position of a particular point with reference to the origin. Using this, you can determine the current position of a body. By understanding the position, you will be able to evaluate the body’s motion. 


As per the displacement vector definition, it is one of the crucial concepts used in Mathematics. It is normally a vector and shows the total direction as well as distance travelled by an object in a straight line. The word displacement vector is used in Physics to explain the acceleration, speed as well as the distance of a particular object moving in a direction. It may be noted that displacement is not a scalar quantity; it is a vector quantity. 


Example of Displacement Vector

Now you all know that a change in a position vector of an object is called the displacement vector. Suppose an object is at the point P at time= 0 and another point Q at time= t. Here, the position vectors at the points P and Q can be represented in the following manner: 

Position vector at point P = $\overrightarrow{P}$ 

Position vector at point Q = $\overrightarrow{Q}$


On the other hand, the displacement vector of a particular object that is travelling from time interval 0 to t can be written as  $\overrightarrow{Q}-\overrightarrow{P}$

The displacement of the object is sometimes defined as a vector distance between the starting point and the endpoint. It may be noted that displacement is not a scalar quantity; it is a vector quantity. 


The Formula of Position Vector

Well, if you know the position of a point in the xy-plane, then you can use a simple formula to know the position vector between those two points. Here, the position vector formula will be 

$\overrightarrow{CB}=\left( {{x}_{2}}-{{x}_{1}} \right)\widehat{i}+\left( {{y}_{2}}-{{y}_{1}} \right)\widehat{j}$ where $B=\left( {{x}_{1}},{{y}_{1}} \right)$ and $C=\left( {{x}_{2}},{{y}_{2}} \right)$. 


Use this formula, and you can ascertain the value accurately. 


How to Find a Position Vector? 

Here is a position vector example that will help you to understand this better. To determine a point’s position vector, you should first determine that particular point’s coordinates. 


Suppose you have two points, i.e., P and K.  Here, $P=\left( {{x}_{1}},{{y}_{1}} \right)$ and $K=\left( {{x}_{2}},{{y}_{2}} \right)$. Then you need to find the vector PK, the position vector between points P and K. To calculate the position vector here, you will have to subtract the P’s corresponding components from K. Here, the equation will be: 

$KP=\left( {{x}_{2}}-{{x}_{1}} \right),\left( {{y}_{2}}-{{y}_{1}} \right)$


Is There Any Difference between a Unit Vector or Position Vector?

As discussed above, the vector is considered to be a unit vector, and it can be used to determine the direction. Besides, it has a magnitude value that is equal to 1. Well, there is no magnitude needed for a direction. That’s why a unit vector’s magnitude will always be equal to 1. The position vector shows the location or position of a given point while considering different arbitrary reference points such as the origin.


Understanding Scalar and Vectors

Quantities, in Physics, are classified in terms of scalars and vectors. Well, there is a simple difference between these two things. The quantities that have directions along with their magnitude related to them are known as vectors. On the other hand, a scalar quantity is a magnitude. In scalar quantities, different arithmetic operations, such as multiplication, subtraction, or addition, are carried out in the same manner as you normally do with the real numbers. For instance, the total of two different scalar quantities with values 0.2 and 0.4 will be 0.6. These rules cannot be used for vector quantities. 


Conclusion

One of the most crucial aspects of kinematics is the displacement vector and position vector. Besides, while understanding the kinematics, you will have to understand the major difference between position and displacement vector. And you can learn more about this from the above sections. In simple words, the position vectors talk about the position of a known object. By knowing the exact position of the object, it will be much easier for you to describe the motion of the object. On the other hand, the variation or change in the position vector means the displacement vector. Use the displacement vector formula mentioned above to calculate the value. 

FAQs on Position Vector and Displacement Vector - JEE

1. Does position matter for vectors?

Yes, the position of vectors plays a vital role for any vector components. The position vector is a sought-after quantity as this talks about the current motion of a particle, known as the point mass. In kinematics, the position vector is a main tool used to determine the direction of the vector as well.

2. When is Vector a Tensor?

Basically, tensors are considered as generalised forms of scalars and vectors in linear vector space or linear algebra. So when we have a vector of rank 1, it will be considered as a tensor. Similarly, scalar is a tensor of rank 0. When we study linear algebra, the concept of tensors is very important.

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