- Vector Algebra Definition: Vector algebra is the branch of mathematics dealing with vectors, quantities that have both magnitude and direction.
- Vector Diagrams: Vector diagrams are visual tools that show the magnitude and direction of vectors, aiding in understanding their relationships.
- Vector Components: A vector can be split into two perpendicular components, typically along the x and y axes.
- Complex Representation: Vectors can be represented using complex numbers, where the imaginary unit ‘j’ indicates a 90-degree rotation.
- Forms of Vectors: Vectors can be represented in different forms: rectangular, complex, trigonometric, and exponential.
Before studying electrical engineering, it’s important to understand the angular relationship between voltage and current. To grasp this, we need to know what a vector is and explore vector algebra and vector diagram.
Definition of Vector
There are some quantities which have both magnitude and as well as their direction of action. This type of quantities is called vector quantity. This is how one can make basic definition of vector in very few words. The most basic concept of a vector is that, it is a representation of these type of quantities in both magnitude and direction. Whenever we represent any quantity it may have some direction of action. Suppose if we say, a force of 5 N, it does not complete the picture.
We must specify the direction of a force, like whether a 5 N force is upward, downward, or in another direction. Thus, a vector quantity includes both magnitude and direction. The direction is represented by the angle between the quantity’s direction and a reference axis.
In this vector diagram, vector OB has a magnitude of |Z| at an angle θ to the reference axis OX. It can be resolved into two perpendicular components.
This is the conventional method of representing a vector.
Vector Algebra
Now, let’s discuss vector algebra. For various calculations, vectors must be expressed algebraically. In the vector diagram, vector Z is the result of adding its components X and Y.
This vector may be written in vector algebra as 
Where, j indicates that the component Y is in perpendicular to component X. The x axis in the vector diagram is known as ‘real’ or ‘in-phase’ axis and the vertical y axis is called ‘imaginary’ or ‘quadrature’ axis. The symbol ‘j’ which is associated with the quadrature component Y, may be considered as an operator which rotates a vector anticlockwise through 90o. If a vector has to be rotated anticlockwise through 180o then the operator j has to perform its function twice and since the vector has reversed its sense then j.j or j2 = − 1
| Which implies, j = √ | − 1 |
So we have seen that a vector quantity can be represented in following different forms,
Relation between rectangular and complex form of a vector
As per the vector diagram shown on this page. The magnitude of vector Z is
From these two equations, we get,
Putting these values of X and Y, in complex form of Z, we get,
The value of above expression is known as trigonometrical form of vector. Again we know that, cosθ and sinθ can be represented in exponential form as follows
If we put these above exponential form of sinθ and cosθ in the equation Z = |Z|(cosθ + jsinθ) we get,
⇒ Z = |Z|ejθ
This is the exponential form of vector.
Therefore from all above expressions of vector algebra and vector diagrams, it can be concluded that a vector quantity can be represented as total four basic form as listed below
