A Short Note on Vectors

Now let us first understand what vectors are? Quantities that have both magnitude and direction are known as vectors in mathematics. The vector’s size is determined by its magnitude. Using a line and an arrow, the magnitude of the vector is shown, and the arrow shows its direction. It is claimed that two vectors are equal if the magnitude and direction of the two vectors are the same. Both engineering and the sciences depend heavily on it. 

We’ve all heard the terms vector and scalar used in conjunction with one another. A scalar quantity has magnitude but no direction.

How to Represent a Vector?

In order to distinguish between magnitude and direction, a directed line segment is marked by the letter AB or simply by the letter a. A directed line segment is denoted by the letter a. Point A, from which the vector AB originates, is referred to as the vector’s beginning point. Point B, from which it finishes, is referred to as the vector’s terminal point in this instance.

Types of Vectors

Vectors are often used in mathematics and are classified into ten main forms. The ten kinds of vectors are as follows:

• Equal vectors

If the magnitudes and directions of the vectors are the same regardless of where their starting points are located, the two vectors are said to be equal.

• Coplanar vectors

Coplanar vectors are three or more vectors that lie in the same plane or are parallel to the plane.

• Zero or Null vectors

When the magnitude of a vector is 0 and the beginning and ending points are the same, it is called a Zero Vector. Line segment AB, for example, has the same coordinates as points A and B. This vector has no particular direction.

• Like and Unlike vectors

Vectors are like when they are moving in the same way, unlike when they’re moving in an opposing direction.

• Unit vector

Essentially, a unit vector is any vector with a magnitude equal to one, and it is used to indicate the direction of any other vector. A cap (^) is used to represent it symbolically.

• Collinear vectors

When two or more vectors are parallel to the same line, they are said to be collinear, regardless of their magnitudes and orientations.

• Position vectors

A position vector represents a point’s location in the three-dimensional Cartesian system about a reference point.

• Negative vectors 

The term “negative vector” refers to a vector with the same magnitude as the provided vector but points opposite.

• Co-initial vectors

Coinitial vectors are vectors that have the same starting point as each other.

• Displacement vectors

If a point is moved from location A to B, the vector AB denotes the displacement vector of the point.

Operations on a Vector

Addition of Vectors

The vector addition operation is accomplished by applying the triangle and parallelogram law of vector addition to any two vectors. Let’s study the two laws in detail:

• Triangle Law

Suppose that two vectors are expressed in direction and magnitude by the two adjoining sides of the triangle taken in sequence and that their result is the closing side of the triangle taken in the opposite order. The triangle rule of vector addition is used to describe this phenomenon.

• Parallelogram Law

Two vectors operating at a point are represented by the two adjoining sides of a parallelogram in magnitude and direction, and the resultant is depicted in direction and magnitude by the horizontal crossing through their common tails. This is a well-known mathematical concept known as the parallelogram rule of vector addition. 

Subtraction of vectors

In the case of any two vectors, p and q, the difference between their vectors p – q are defined as p +(- q).

Multiplication of Vectors

To multiply two vectors, you must determine their “cross product” or “dot product.”

• Dot product

When two vector values are added together, they produce a scalar quantity called the dot product. If two vectors “a” and “b” have the same magnitude, their dot product is “a. b,” which is produced by multiplying the magnitude by the cosine of the angles. As a result, it may be written as A. B = IAI IBI Cos θ.θ.

• Cross product

The cross product is the product of two vectors that results in the production of a vector quantity. A cross product is indicated by ” a × b.” It is created by multiplying the magnitude of the angles by their sine, multiplied by a unit vector, i.e., “n.” Thus, the cross product is defined as A x B = IAI IBI Sin θ n.

Conclusion 

In many areas of mathematics and science, vectors and scalars are significant. Essentially, vectors may be used to represent any physical phenomenon that has both a magnitude and a direction, and they are particularly useful in mathematics. Using them to describe the displacement or velocity of a moving item is helpful in situations when a single number would not be sufficient to describe the object’s movement. There are several real-world applications for vector analysis, making it an important topic of research.