CONTENTS

S.N. Topic Page no.

1. Objectives 4
2. Introduction to Vector 5
3. Vector 5-9
4. Application of Vector 11

2 |Introduction to Vector
 INTRODUCTION TO VECTOR
Vectors are mathematical objects that represent quantities with both magnitude and
direction. They are fundamental tools used in various fields such as physics,
mathematics, engineering, and computer science. Vectors provide a concise and
powerful way to describe and analyze physical phenomena and mathematical
concepts.

In a geometric sense, vectors can be visualized as arrows in space. The length of the
arrow represents the magnitude or size of the vector, while the direction of the arrow
indicates its orientation. Vectors can exist in two-dimensional space (having two
components) or three-dimensional space (having three components), but they can also
be extended to higher dimensions.

Understanding vectors and their properties is essential for tackling problems involving
physical systems, motion, and mathematical modeling. They provide a powerful tool
set for analyzing and solving complex problems with both magnitude and direction,
making them a cornerstone of mathematical and scientific thinking.

VECTOR
The vectors are defined as an object containing both magnitude and direction. Vector
describes the movement of an object from one point to another. Vector math can be
geometrically pictured by the directed line segment (a). They are used in linear
algebra to represent points, displacements, velocities, forces, and other quantities.
The length of the segment of the directed line is called the magnitude of a vector and
the angle at which the vector is inclined shows the direction of the vector. The starting
point of a vector is called “Tail” and the ending point (having an arrow) is called
“Head.”
A vector is defined as a mathematical structure. It has many applications in the field
of physics and geometry. We know that the location of the points on the coordinate
plane can be represented using the ordered pair such as (x, y). The usage of the vector
is very useful in the simplification process of three-dimensional geometry.
Along with the term vector, we have heard the term scalar. A scalar actually
represents the “real numbers”. In simpler words, a vector of “n” dimensions is an
ordered collection of n elements called “components“.

Examples of Vectors
The most common examples of the vector are Velocity, Acceleration, Force,
Increase/Decrease in Temperature etc. All these quantities have directions and
magnitude both. Therefore, it is necessary to calculate them in their vector form.
Also, speed is a quantity that has magnitude but no direction. This is the basic
difference between speed and velocity.

4 |Introduction to Vector
Vector Notation
As we know already, a vector has both magnitude and direction. In the above figure,
the length of the line AB is the magnitude and head of the arrow points towards the
direction.

Therefore, vectors between two points A and B is given as AB, or vector a.

Arrow over the head of the vector shows the direction of vector.

Magnitude of a Vector
The magnitude of a vector is shown by vertical lines on both the sides of the given vector “|
a|”. It represents the length of the vector. Mathematically, the magnitude of a vector is
calculated by the help of “Pythagoras Theorem,” i.e.
|a|= √(x2+y2)

Unit Vector
A unit vector has a length (or magnitude) equal to one, which is basically used to
show the direction of any vector. A unit vector is equal to the ratio of a vector and its
magnitude. Symbolically, it is represented by a cap or hat (^).
If a is vector of arbitrary length and its magnitude is ||a||, then the unit vector is given
by:

It is also known as normalizing a vector.

Zero Vector
A vector with zero magnitudes is called a zero vector. The coordinates of zero vector
are given by (0,0,0) and it is usually represented by 0 with an arrow (→) at the top or
just 0.
The sum of any vector with zero vector is equal to the vector itself, i.e., if ‘a’ is any
vector, then;
0+a = a
Note: There is no unit vector for zero vector and it cannot be normalised.
Operations on Vectors
5 |Introduction to Vector
In maths, we have learned the different operations we perform on numbers. Let us
learn here the vector operation such as Addition, Subtraction, Multiplication on
vectors.
Addition of Vectors
The two vectors a and b can be added giving the sum to be a + b. This requires joining
them head to tail.

We can translate the vector b till its tail meets the head of a. The line segment that is
directed from the tail of vector a to the head of vector b is the vector “a + b”.

Characteristics of Vector Math Addition

 Commutative Law- the order of addition does not matter, i.e, a + b = b + a

 Associative law- the sum of three vectors has nothing to do with which pair
of the vectors are added at the beginning.
i.e. (a + b) + c = a + (b + c)
Subtraction of Vectors
Before going to the operation it is necessary to know about the reverse vector(-a).

A reverse vector (-a) which is opposite of ‘a’ has a similar magnitude as ‘a’ but
pointed in the opposite direction.
First, we find the reverse vector.
Then add them as the usual addition.
Such as if we want to find vector b – a

6 |Introduction to Vector
Then, b – a = b + (-a)

Scalar Multiplication of Vectors

Multiplication of a vector by a scalar quantity is called “Scaling.” In this type of
multiplication, only the magnitude of a vector is changed not the direction.

 S(a+b) = Sa + Sb
 (S+T)a = Sa + Ta
 a.1 = a
 a.0 = 0
 a.(-1) = -a

Scalar Triple Product

The scalar triple product, also called a box product or mixed triple product, of three
vectors, say a, b and c is given by (a×b)⋅c. Since it involves dot product and evaluates
single value, therefore stated as the scalar product. It is also denoted by (a b c).
(a b c) = (a×b)⋅c

The major application of the scalar triple product can be seen while determining
the volume of a parallelepiped, which is equal to the absolute value of |(a×b)⋅c|,
where a, b and c are the vectors denoting the sides of parallelepiped respectively.
Hence,
Volume of parallelepiped = ∥a×b∥ ∥c∥ |cosϕ|=|(a×b)⋅c|

Vector Multiplication
Basically, there are two types of vector multiplication:

 Cross product
 Dot product

Cross Product of Vectors

The cross product of two vectors results in a vector quantity. It is represented by a
cross sign between two vectors.
i.e., a × b

7 |Introduction to Vector
 The mathematical value of a cross product-

where,
| a | is the magnitude of vector a.
| b | is the magnitude of vector b.
θ is the angle between two vectors a & b.

Dot product of Vectors

The dot product of two vectors always results in scalar quantity, i.e. it has only
magnitude and no direction. It is represented by a dot (.) in between two vectors.
a dot b = a. b
The mathematical value of the dot product is given as
a . b = | a | | b | cos θ

Components of Vectors (Horizontal & Vertical)

There are two components of a vector in the x-y plane.

1. Horizontal Component
2. Vertical Component
Breaking a vector into its x and y components in the vector space is the most common
way for solving vectors.

A vector “a” is inclined with horizontal having an angle equal to θ.

This given vector “a” can be broken down into two components i.e. ax and ay.
The component ax is called a “Horizontal component” whose value is a cos θ.
The component ay is called a “Vertical component” whose value is a sin θ.
8 |Introduction to Vector
 APPLICATION OF VECTOR
Vectors can be added, subtracted, and scaled, allowing for the formulation of
mathematical operations and relationships. The importance of vectors extends beyond
mathematics and into various fields:

1. Physics: Vectors are fundamental in describing physical phenomena such as

motion, forces, and fields. Velocity, acceleration, and momentum are all vector
quantities. In physics, vector calculus is used to study electromagnetism, fluid
dynamics, and other areas.

2. Computer Graphics: Vectors are used extensively in computer graphics to

represent geometric entities such as points, lines, and shapes. They enable
transformations, rotations, and scaling of objects in three-dimensional space.

3. Engineering: Vectors are used in engineering disciplines, such as civil,

mechanical, and electrical engineering. They are employed to represent forces,
moments, velocities, and other physical quantities. Vector analysis is essential in
fields like structural analysis, fluid mechanics, and circuit analysis.

4. Robotics: Vectors are crucial in robotics for modeling and controlling the motion
of robotic systems. They describe the position and orientation of objects, the
movement of robot arms, and the forces exerted by actuators.

5. Computer Science: Vectors are employed in various algorithms and data

structures. They are used in machine learning for representing and manipulating
features and data points. Vector spaces and linear transformations are fundamental
concepts in computational mathematics.

9 |Application of Vector
 EXAMPLES OF APPLICATION OF VECTOR
IN REAL LIFE

Vectors in Video Games

In video games, we use vectors to represent the velocity of players, but also to control
where they are aiming, or what they can see (where they are facing). All of this with
one vector. We also need a point to keep track of the player’s position at all times.

In this case, the player’s position will be the origin for our velocity and rotation vector.

Let’s say we have an enemy AI that needs to shoot all the boxes he encounters.
Imagine the AI has a velocity and direction vector of (1,0) and he’s standing at the
origin of the map, so (0,0). He finds a box at position (2,3), so we want to rotate him
so he can point at the box and shoot it, but we need to calculate the angle between
where he’s aiming, and where he wants to actually aim to hit the box.

Vector in computer programming

In computer programming, a vector is either a pointer or an array with only one

dimension. A vector is often represented as a 1-dimensional array of numbers, referred
to as components and is displayed either in column form or row form.Vectors are a
logical element in programming languages that are used for storing data. In computer
graphics, the term vector describes a line with a starting and ending point.

10 |Application of
 CONCLUSION

In conclusion, the project on vectors has provided a comprehensive exploration and

application of vector concepts and operations. Throughout the project, we have gained
a deep understanding of vectors, their properties, and their significance in various
mathematical disciplines.By applying vector operations, such as addition, subtraction,
scalar multiplication, dot product, and cross product, we have successfully solved
mathematical problems and tackled real-world scenarios across different domains. The
project has showcased the versatility and power of vectors in modeling and solving
complex problems in geometry, calculus, linear algebra, physics, and beyond.
Furthermore, the project has enhanced our critical thinking skills by challenging us to
analyze and interpret mathematical results, evaluate different solution methods, and
assess the limitations and strengths of vector-based approaches.
In summary, the project on vectors has not only deepened our understanding of vector
concepts but has also provided us with a robust toolkit for problem-solving in various
mathematical and real-world contexts.

11 |Application of
REFERENCES
1. Byjus.com

2. Google

3. https://medium.com/analytics-vidhya/what-is-a-vector-5c86fc2b57c1

12 |Application of