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Vector Algebra

What do you understand by ‘vector algebra'? What are the basics of 'vector algebra’? What do we mean by vector and matrix algebra? We will be shedding light on all these topics in an interesting and easy way.

In physics and engineering, vector algebra plays a notable role in several applications such as addition and multiplication across various physical quantities represented in the form of vectors in three-dimensional spaces. In simple words, vector algebra is used for performing diverse algebraic operations which involve vectors. 

Vector

The word vector is a Latin word that means ‘carrier’. Vectors carry a point X to Y. The length of the line from point X to Y is known as the magnitude of the vector and the direction of displacement of point to point Y will be known as the direction of the vector XY. Other names for vectors are Spatial vectors and Euclidean vectors. In various fields such as physics, maths, engineering, vectors are used. 

Vector and matrix algebra

Vectors and matrices refer to an array of numerical values. The algebra for symbolic operations on them is not similar to the algebraic operations on scalar or single numerical. In other words, one can also say that the algebraic matrix notation is shorthand for the immediate corresponding scalar longhand. A vector can be described as a number column whereas a matrix can be described as a rectangular table of numbers. Vector is an array of those numbers wherein the order of numbers is also of importance. Mostly these can be represented by a lowercase bold letter. On the other hand, a matric involved a two-dimensional array of numbers. In general, matrices can be represented by an uppercase bold letter such as B. 

A vector with some particular dimension such as ‘m’ can be looked at as a matrix with ‘m’ rows and one column. A matrix with dimensions m*n can be viewed as being composed of ‘m’ column vectors or ‘n’ rows vectors. 

Types of Vectors

For diverse algebraic expressions, different types of vectors can be used. The different types of vectors are termed depending on their direction, magnitude and relationship with other vectors. Here are some types of vectors along with their properties – 

  1. Zero Vectors – These are vectors having zero magnitudes. The vector with zero magnitude does not have a direction either. Zero vectors can also be called the additive identity of vectors.
  2. Unit Vectors – Those vectors which have a magnitude equal to one are known as unit vectors. Unit vectors are also known as the multiplicative identity of vectors. The unit vectors have a magnitude of one. 
  3. Position Vectors – Those vectors which are used for determining the position and direction of movement of vectors in a three-dimensional space can be called position vectors. In position vectors, their magnitude, as well as direction, can get changed relative to other bodies. Position vectors are also known as location vectors. 
  4. Equal Vectors – If the corresponding components of two or more than two vectors are equal then they are known as equal vectors. These vectors have the same magnitude and direction. These vectors might have differing initial as well as terminal points however the direction and magnitude must be equal.
  5. Negative Vectors – The vectors in which the magnitude is the same but the direction differs are known as negative vectors. 
  6. Parallel Vectors – If two or more than two vectors have the same direction but the magnitude may differ then they are known as parallel vectors. 
  7. Co-initial vectors – Those vectors which have the same initial point are known as co-initial vectors. 

Conclusion

As we have come to the conclusion of this topic, the concept around vectors and matrices has become clear. Vectors can be understood as the carrier. In diverse fields such as physics, maths, engineering, vectors are commonly used. Vector algebra is used for performing diverse algebraic operations which involve vectors. A vector can be explained as a column of numbers whereas a matrix can be explained as a rectangular table of numbers. There are various different types of vectors that can be used for diverse algebraic expressions. The diverse types of vectors are mainly based on depending on their direction, magnitude and relationship with other vectors. The types of vectors include zero vectors, equal vectors, negative vectors, parallel vectors, co-initial vectors, unit vectors and position vectors.