Vector

The physical quantity that shows both magnitude and direction is a vector quantity. 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. 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. 

Types of vectors

There are numerous types in which we can find and describe the vector. Each of them is 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. 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.Equal Vectors

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.  

Product of Two Vectors

Dot Product

The dot product of vectors is the result of the magnitude of the two vectors and the cos angle between them. For example, if the magnitude of Vector A and B is |A||B| and the angle between the vectors is cosθ, then the dot product becomes A.B= |A||B|cosθ.

This formula represents the magnitude and the angle together, where (A.B) is the dot product, |A| is the magnitude of vector A, |B| is the magnitude of the vector of B and cosθ is the cosine angle in the middle of the two vectors. In some scenarios, the value of the dot product might be nil.

Vector products 

The Vector product of the two vectors is the result of the magnitude of the two vectors and the sinθ angle between both vectors. For example, if the two vectors’ value is A and B, the magnitude of the vectors is |A||B|, and the sine angle between them is sinθ. Therefore, the formula of the cross-products between two vectors is AxB=|A||B|Sinθ.

Right-Hand Vector product Rule

The right-hand Vector product rule highlights the direction of the vector. This rule states that we must stretch the index finger of the right hand towards the direction of the first vector (A) and the middle finger towards the second vector (B). As a result, the hand’s thumb will show the direction of the cross product (AxB). 

Conclusion

Vectors and scalars are concepts in mathematics that may be difficult to understand at first. On the other hand, with consistent study and comprehension, the knowledge becomes manageable. In a nutshell, a scalar quantity is a representation of the magnitude of a quantity. However, the vector quantity reveals the amount of an item and the direction. Thus, the fundamental distinction between scalar and vector values is one of direction; scalar quantities lack direction, but vector numbers do.