General Vectors and Notation

The physical quantity that shows both magnitude and direction is a vector quantity. They have initial points and terminal points and are represented by arrows. The advancement of the algebra of vectors and vector analysis as we know it was first revealed in sets of remarkable notes made by J. Willard Gibbs (1839-1903), also known as the Father of Vector Algebra. His notes were intended for his students at Yale University. 

Vectors

The vector is the quantity that combines the duo – magnitude and direction. Vectors are depicted by the direct pointed line in which the length shows the vector and magnitude and the orientation shows the direction of the vector. 

Types of vectors

There are numerous types in which we can find and describe the vector. Each of them are unique, though easy to understand. They have certain properties to describe them. 

1.  Co-initial Vector

 Co-initial vectors are a form of vector in which the beginning points of two or more distinct vectors are the same. All vectors in this sort of vector begin from the same point. The vectors are called co-initial vectors because their origin points are the same. 

2.  Collinear Vectors

The collinear vector is another sort of vector in which two or more vectors, regardless of magnitude or direction, are parallel. Because they are parallel, they never cross paths.

3.  Zero Vectors

The zero vector is another sort of vector in which the value of the vector is zero and also the origin and endpoint points of the vector are the same. The zero vector has no constituents and will not point in any direction.

4.  Unit Vector

The unit vector is a subtype of a vector with a value based on the length of one unit. Unit vectors are defined as any vectors with a magnitude of one.

5.  Position Vector

A position vector is a vector in which the origin point is set to O and one random point in the space is designated as A. The position vector with the reference origin O is then known as vector AO. 

6.  Coplanar Vectors

Coplanar vectors have three or more vectors in the same plane or that can lie in the parallel plane. There is always the chance of finding any two different vectors in the same plane, referred to as coplanar vectors.

7.  Unlike and Like Vectors 

Like vectors are the type of vectors that have the same direction and are referred to as such. Unlike vectors are vectors that have the same direction but are in opposite directions. 

8.  Equality of vector

Equal vectors are the sort of vector in which two or more vectors with the same magnitude and direction are considered equal.

9. Displacement Vector

The displacement vector is the sort of vector that occurs when one vector is shifted from its original position. The vector distance between the object’s starting and final points can determine the displacement.

10. Negative Vector

A negative vector is a form of vector in which the value of both vectors is equal, but the direction of both vectors is opposite. Then we can write them as

 a = -b. This is known as a negative vector.  

Vector Addition

A vector represents direction and magnitude. So, it is usually represented with an arrow above the quantity. For example, Vector a is written as a and so on. When you add two vectors, the resultant is written as a + b. 

There are some basic principles to keep in mind for addition of vectors: 

• Vector addition is only possible when the vectors are of the same nature. For instance, you can add acceleration only with acceleration and not any other quantity. 

• Scalars and vectors cannot be added. 

Methods of vector addition

Some of the other laws to consider before attempting addition and subtraction of vectors are:-

Triangle Law

Adding two vectors represented by a  and b can be done by drawing a line AB where A forms the tail and B is the head.

• Draw a second line BC where B forms the tail and C its head

• Join the points A and C by another line. The resulting line AC will give you the sum of the two vectors.

Parallelogram Law

The addition and subtraction of vectors, particularly the procedure of addition, can be explained with a parallelogram too. The law elaborates the process in the following manner:-

• P and Q represent two sides of a parallelogram that are adjacent to each other. Each side points in an external direction.

• A diagonal line drawn from the intersecting point of P and Q will give the result.

Polygon Law

This particular law considers multiple counts of vectors. When their magnitude along with the direction of each is considered in an order of the sides, the result is evident by the magnitude and the direction considered in the opposite direction of its sides. The addition and Subtraction of vectors by using a polygon can be depicted by the following formula:-

R =A+B+C+D+……….

Here A, B, C….. are sides of polygon and R is the resultant vector to the polygon.

Notation of vectors

A vector is an object usually represented as an arrow over a letter. ‘Vector’ is a Latin word that means carrier. Physically, vectors are represented by directed line segments, each with an arrow denoting the vector’s direction and a length equal to the vector’s magnitude. The vector’s axis travels from tail to head.

Conclusion :

In mathematics and physics, the vector is a quantity that has direction and magnitude. There are several different types of vectors. These are zero vector, unit vector, coinitial vectors, equal vectors and negative of a vector. In physics and mathematics, scalar has only magnitude and no direction. The scalar quantity is denoted as ,while the vector is denoted with a the product of both will be a.