[Maths Class Notes] on Vector and Scalar Quantities Pdf for Exam

Humans invented Math and Science so that they can better understand and describe the world they live in. This world is like a four-dimensional figure which is governed by the passing of time. It is also like a three-dimensional space i.e., up and down, left and right, and back and forth. If we sincerely notice, we’ll see there are two types of quantities and processes in our world. One that depends on the direction in which they occur and the other that does not depend on direction. For example, the volume of an object is the three-dimensional space that it occupies. But does it have anything to do with direction? No, it is not related to the direction in any way. A  5 cubic foot block of iron will be the same if we move it up and down then left and right but the location of an object will always be dependent on the direction because If a 5 cubic foot block is moved to 5 miles towards the north, the resulting location will be very different if we moved it 5 miles to the east. Thus, mathematicians and scientists named this quantity as ‘vector’ if it depends on the direction. They call it ‘scalar’ if the quantity does not depend on direction.

Scalar Quantity Definition

Physical quantities that can be fully described by giving a single number and an appropriate unit are called a Scalar quantities. Scalar is a synonym of number. For example, if Sam is driving a car at 65 miles per hour, the only information that can be drawn is a scalar measurement speed i.e., 65 miles per hour. But if we are informed that his car is moving southwest at  65 miles per hour then the information is a vector quantity as it also tells us the direction of the car too. Hence, scalar quantities are described completely based on magnitude.

Few other examples of scalar quantities are “an interview lasts 50 min” or “a gas tank of a car can hold 65 L” or “the distance between two poles is 100 m.”

Vector Quantity Definition

We have spoken and understood what a scalar quantity is. So now let’s learn what a vector quantity means. Vector representation can be done as a physical quantity that cannot be fully described by a single number of physical units. For example, if you want to go to a place you’ve never been to, you not only need to know the distance you have to travel but also the direction of your destination. Hence, vector quantities are described based on the magnitude as well as direction. If you are studying motion, knowing about vectors is very crucial. Vectors allow complex multi-dimensional problems to be viewed as one-dimensional problems.

Scalar Quantity Examples:

Here are a few scalar quantity examples that will help you to understand the concept better. 

• There is no direction in mass but only value. Thus, it is a scalar quantity.

• Time, distance, temperature, length, volume, and energy are also examples of scalar quantities.

• The electric charge also has only value and no direction. Thus, this also comes under scalar quantity. 

• Some examples of scalar quantities also include speed and power.

Vector Quantity Examples:

Here are a few vector quantity examples that will help you to understand the concept better. 

• Force has a value and a direction therefore the basic example of vector quantity can be a pull or a push. When we pull or push something, we apply strength i.e., force (magnitude) but it always includes a direction. 

• Weight too not only has value but a direction. Your weight is directly proportional to your mass (magnitude) which is always in the direction towards the centre of the earth.

• Some other examples of vector quantities include velocity, acceleration, displacement, and momentum.

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Addition of Vector 

The addition of two vectors can be defined as the diagonal of a parallelogram or law of parallelogram. According to this law, when two vectors say A and B are placed at the same point represented in magnitude as well as direction, the result will be in magnitude and direction by the diagonal of the parallelogram that passes through the same point.

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Subtraction of Vector

Subtraction of one vector from another is to be done in the same way as that of addition. The only difference will be that we will add the negative of the subtracting vector as shown in the picture.

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Difference between Scalar and Vector Quantity

Notation

There is a difference between scalar and vector quantity hence, the vector has its notation. The symbol of a vector can be written in many ways. One of them can be an arrow pointing to the right, just above it.

[vec{F}] represents the force vector

F represents the magnitude of the force vector

A vector is represented as an arrow having a head and a tail. The length of the arrow is the representation of the magnitude of the vector.

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Few interesting Facts About Scalars and Vectors

• An Irish physicist, William Rowan Hamilton is credited for inventing vectors.

• Vectors were developed in their modern form by Josiah Willard Gibbs and Oliver Heaviside. They applied it to express the laws of electromagnetism that were postulated by James Clerk Maxwell.

• Vectors and scalars are very crucial in many fields of Mathematics and Science.

• Vectors can be represented in two dimensional or three-dimensional space.

• Vector graphics are sometimes used in computers. Vectors can be scaled to a larger size without even losing any image quality.

• Cross product of two vectors results in another vector whose direction is orth
  ogonal concerning both the original vectors. It is illustrated by the right-hand thumb rule which you often study in Physics.  

Conclusion

The topic of Vector And Scalar Quantities forms a base for many topics of Physics. Thus you can understand how important it is not just for mathematics but for other subjects as well. You may also realise how important this topic is for your entrance exams after going through the previous year papers. 

Many times students find this subject difficult due to the various theorems involved and how it is slightly different from traditional mathematics. understands this therefore, it has ensured that all the content made available to you are in the best possible format so that you don’t have to go around looking for each topic separately.

To make this more bearable and interesting for you we have added interesting facts below. This will help in retaining your interest as well as encourage your curiosity towards the subject.