Vectors

Vector arrows depicting magnitude are the foundation of logical statements in mathematics. There are certain types of vectors; however, all vectors are categorised into two types: unit vector and zero Vector. Any form of vector with no magnitude and coordinates is categorised as zero vectors, and the remaining ones are unit vectors. 

Vectors have associative and cumulative properties, which are used in additions. Apart from that, they are also used in different operations. Since it is such a broad concept, here’s detailed study material covering all aspects of vectors.

About Vectors

In a nutshell, those objects which have directional magnitude can be termed vectors. They are represented by a line with an arrow at one end that points to the direction of the object. The length of the line depicts the magnitude. Therefore, vectors can be shown by arrows that point in a particular direction and have an initiating and terminating point. They are also known as Euclidean vectors or spatial vectors.

Vectors were used by scholars from as early as the 19th century to study the electromagnetic field. The physical quantities like acceleration, velocity, displacement, etc., are represented by the vectors, which show that these quantities have both magnitude and direction.

Representing the Vectors

The representation of the vectors is done in bold lowercase letters like ‘a’ or by using an arrow over it. The direction of the arrow is in the direction of the vector. The vectors are also denoted as the initial and terminal points with arrows over them. For example, vector AB can also be written as 

The equations a, b, and c are real numbers, and i, j, k are unit vectors in all the three-axis- x, y, and z. Two vectors are considered the same if they have the same direction and magnitude.

The Magnitude of the Vector 

The magnitude of the vectors is defined as the length of the vector. For example, a vector AB has a magnitude of length between A-B. It can be represented as two vertical lines on both sides of the vector AB like |AB|. The length of the vector in mathematics can be calculated through the Pythagoras theorem as follows:

               |AB| = √(x2+y2)

For example, force and velocity are both vector quantities, and the magnitude of the vectors can be defined as the strength of the force and what speed is associated with the velocity.

Types of Vectors 

Vectors can be of many types depending on the magnitude, direction, or their relation with another vector. Two main types of vectors are unit vector and zero vector. 

Unit Vector – A vector with a magnitude of length equal to one is known as a unit vector. The ratio between vectors and their magnitude is a unit vector. It is denoted by the symbol (^). They are also known as the multiplicative identity of the vectors. 

Zero Vector – The vectors having zero magnitudes are called zero vectors. They have 0 coordinates and are denoted as (0) or an arrow over zero. They are also called the additive identity of the vectors. 

Co-initial vectors, like and unlike, coplanar, collinear, and positional are some other types of vectors. However, they are also categorised as unit and zero vectors.

Operations on Vectors 

We will now briefly study the various operations on vectors geometrically: addition, subtraction, and multiplication, both with scalar and vector. 

Addition of Vectors 

Two vectors, a and b, can be added to get  a+b and can describe it as the tail of one coincides with the tail of the other. Hence, the resulting vector can be mentioned as vector ‘a+b’. There are two characteristics associated with vectors:

• Commutative Property – The order of the vectors can be shuffled; it does not change the value.

                   a+b = b+a

• Associative Property – If three vectors are added, any pair can be added first; it does not change the value. 

            (a+b)+ c = a+(b+c)

Subtraction of Vectors 

The subtraction of vectors is similar to addition. But the direction of one vector is changed, and hence, the sign (-) is added to it. The vector has an opposite direction, but the same magnitude is called a reverse vector. Vector (-a ) is a reverse vector to (a).

           b – a = b + (-a)

Multiplication of Vectors 

• Scalar Multiplication – Any real number having no direction is known as a scalar. The multiplication of a scalar with a vector can be done by multiplying each component of the vector by that scalar.

                i.e., S × (a+b) = Sa + Sb.

• Scalar Triple Product – The box product of three vectors is called the triple product. It can also be described as the dot product of one vector with the cross product of two vectors. The scalar triple product follows the associative property. 

a.(b×c) = b.(c×a) = c.(a×b)

Conclusion

From the above sections, we have learned all the aspects of vectors, including their operations, representations, types, and magnitudes, along with all general information. We can now say that vectors are the magnitudes that move in a sole direction. They are generally represented as ‘a’. Vectors are categorised into two main types: unit and zero. Also, operations such as addition, subtraction, and multiplications are done between two directional magnitudes or, say, vectors.