In Mathematics, a topic like functions is one of the most important topics as this topic serves as a basic building for whole calculus (differential and integral). Therefore, you must know the important functions such as Linear, Constant, Inverse Functions and others for their use in other important mathematics topics asked in various defence and engineering entrance exams.
Vectors is one of the very important topics that need to be covered in the preparation for the aforementioned competitive exams. The study of vector concepts like Addition, Subtraction and Product of Vectors are important as it is not only used in mathematics but also in many applications in Physics to calculate various quantities in maths and physics.
On that note, let’s discuss the concepts of Vectors and Functions with their different types and application using various examples for better understanding.
Vectors
The vectors are defined as an object containing both magnitude and direction. The movement of an object from one point to another is described by a vector.
Vectors between two points A and B are given as AB with an arrow above it, vector AB or vector a. The vector’s direction is shown by the arrow over its head. It has many applications in maths, physics, engineering, and various other fields.
The most common examples of the vector are Velocity, Acceleration, Force, Increase/Decrease in Temperature, etc.
Types of Vectors
Vectors are classified into distinct categories based on their magnitude, direction, and relationship to other vectors. Few types of vectors include Unit Vector, Zero Vector, Position Vectors, Equal Vectors, Negative Vector, Parallel Vectors, Orthogonal Vectors and more.
Important Operations on Vectors
Let us now look at vector operations such as addition, subtraction, and multiplication.
Addition of Vectors
The two vectors a and b can be added to provide the sum a + b. This necessitates connecting them from head to tail.
Subtraction of Vectors
Before beginning the procedure, it is vital to understand the reverse vector (-a).
A reverse vector (-a) has the same magnitude as a vector but is pointing in the opposite direction.
First, we determine the reverse vector.
Then proceed as usual with the addition.
Suppose we want to find vector b – a.
Then, b – a = b + (-a)
Scalar Multiplication of Vectors
Scaling is the multiplication of a vector by a scalar quantity. Only the magnitude of a vector is modified in this sort of multiplication, not its direction.
Cross Product of Vectors
A vector quantity is produced by the cross-product of two vectors. A cross sign between two vectors represents it.
a ×
A cross product’s mathematical value-
a × b= |a||b|sin θ n̂
What is a Function?
A function is a connection that specifies that each input has only one output (or) it is a special type of relation (a set of ordered pairs) that follows a rule, such as each X-value having only one y-value.
Let’s take a look at how a function’s Domain and Range are defined.
Domain: It is a collection of the ordered pair’s first values (Set of all input (x) values).
Range: It is a collection of the ordered pair’s second values (Set of all output (y) values).
Example:
In the relation, {(-1, 3), {7, 5), (6, -5), (-2, 3)},
The domain is {-1, 7, 6, -2} while the range is {-5, 3, 5}.
Types of Functions
Function types can be defined using the following relationships:
One to one function or Injective function
f: P denotes a function. Q is said to be one-to-one if each element of P has a single element of Q. The domain is {-2, 4, 6} while the range is {-5, 3, 5}.
Many to one function
This function connects two or more P elements to a single Q element.
Onto Function or Surjective function
For each element of set Q, this function has a pre-image in set P.
One-one correspondence
P elements correspond to discrete elements in Q, while Q elements correspond to pre-images in P.
Special Functions in Algebra
The following are some of the most important functions:
Identity Function
Constant Function
Inverse Functions
Absolute Value Function
Linear Function
Composite function