Vector Algebra

Course- Field Theory (ENEL2FT) (2023)

Course Lecturer- Dr. P. Kumar
Email- kumarp@ukzn.az.za
1
 Introduction
⇒In electromagnetic field theory course, the concepts of electric and magnetic fields are
studied.
⇒There are many applications of electromagnetic concepts such as Antennas,
microwaves, electrical machines, bio-electromagnetics, nuclear research, optical fibers,
electromechanical energy conversion etc.
⇒The behavior of electric and magnetic fields are governed by some laws. These laws
are denoted by Maxwell’s equations.
⇒ The electric and magnetic fields are vector quantities and in the analysis the
knowledge of vector algebra is required.

2
Fig. EM waves propagation [2]
 Scalars, vectors and fields
⇒If a quantity has only magnitude, is known as scalar.
Examples- time, temperature, distance, electric potential, population etc.
⇒If a quantity is denoted by the magnitude along with direction, is known as
vector.
Examples- velocity, force, displacement, electric field intensity etc.
⇒The scalar quantity is denoted by simply a letter e.g. 𝐴𝐴, 𝐵𝐵, 𝐶𝐶…, and the vector
quantity is generally denoted by letters with an arrow on top e.g.𝐴𝐴, ⃗ or
⃗ 𝐵𝐵, 𝐶𝐶,…
by bold letters.
⇒The field is a function that describes the quantity in a region. Field may be
scalar or vector depends upon the quantity.
Examples- temperature distribution in building, sound intensity in a theater,
electric potential in a region etc.
3
 Unit vector
⇒A vector quantity has the magnitude as well as the direction. If the magnitude of
the vector quantity is unity, it is known as unit vector.
𝐴𝐴⃗
̂ ̂
𝐴𝐴 = , where 𝐴𝐴=unit vector, 𝐴𝐴 =magnitude of 𝐴𝐴.⃗
𝐴𝐴
⃗ 𝑥𝑥 𝑎𝑎
⇒𝐴𝐴=𝐴𝐴 �𝑥𝑥 + 𝐴𝐴𝑦𝑦 𝑎𝑎�𝑦𝑦 +𝐴𝐴𝑧𝑧 𝑎𝑎
�𝑧𝑧 ,
where 𝑎𝑎�𝑥𝑥 , 𝑎𝑎�𝑦𝑦 , 𝑎𝑎
�𝑧𝑧 are the unit vectors along x-axis,
y-axis and z-axis, respectively. 𝐴𝐴𝑥𝑥 , 𝐴𝐴𝑦𝑦 , 𝐴𝐴𝑧𝑧 are
magnitudes of vector 𝐴𝐴⃗ along x-axis, y-axis,
z-axis, respectively.
�𝑥𝑥 +𝐴𝐴𝑦𝑦 𝑎𝑎�𝑦𝑦 +𝐴𝐴𝑧𝑧 𝑎𝑎
𝐴𝐴𝑥𝑥 𝑎𝑎 �𝑧𝑧
̂
⇒𝐴𝐴 =
𝐴𝐴2𝑥𝑥 +𝐴𝐴2𝑦𝑦 +𝐴𝐴2𝑧𝑧

⇒The unit vectors can also be represented by

𝚤𝚤,̂ 𝚥𝚥,̂ 𝑘𝑘� or 𝑥𝑥, � 𝑧𝑧̂ or 𝑖𝑖�1 , 𝑖𝑖�2 , 𝑖𝑖�3 or 𝑖𝑖�𝑥𝑥 , 𝑖𝑖�𝑦𝑦 , 𝑖𝑖�𝑧𝑧 along x-axis,
� 𝑦𝑦,
y-axis and z-axis, respectively.
Fig. Unit vectors and vector representation using unit vectors
4 [1]
 Examples-magnitude of vectors
• If vector 𝐴𝐴⃗ is defined as:
𝐴𝐴⃗ = 4𝑖𝑖�1 + 6𝑖𝑖�2
The magnitude of vector 𝐴𝐴⃗ is given by:

A = 4 2 + 6 2 = 7.211

• If vector 𝐵𝐵 is defined as:

𝐵𝐵 = 4𝑖𝑖�1 + 6𝑖𝑖�2 − 2𝑖𝑖�3
then the magnitude of 𝐵𝐵 is:

B = 4iˆ1 + 6iˆ2 − 2i3 = 4 2 + 6 2 + 2 2 = 7.4833

5
 Vector addition
• Two vectors i.e. 𝐴𝐴⃗ , and 𝐵𝐵 can be added to give new vector 𝐶𝐶⃗ i. e. 𝐶𝐶= ⃗ 𝐴𝐴+𝐵𝐵.
⃗
• The addition rule for two vectors is given below.
⃗ 𝑥𝑥 𝑎𝑎
Suppose 𝐴𝐴=𝐴𝐴 �𝑧𝑧 , and 𝐵𝐵=𝐵𝐵𝑥𝑥 𝑎𝑎
�𝑥𝑥 + 𝐴𝐴𝑦𝑦 𝑎𝑎�𝑦𝑦 + 𝐴𝐴𝑧𝑧 𝑎𝑎 �𝑧𝑧 , then,
�𝑥𝑥 + 𝐵𝐵𝑦𝑦 𝑎𝑎�𝑦𝑦 + 𝐵𝐵𝑧𝑧 𝑎𝑎
⃗ 𝐴𝐴+𝐵𝐵
𝐶𝐶= ⃗ = 𝐴𝐴𝑥𝑥 + 𝐵𝐵𝑥𝑥 𝑎𝑎
�𝑥𝑥 + 𝐴𝐴𝑦𝑦 + 𝐵𝐵𝑦𝑦 𝑎𝑎�𝑦𝑦 + 𝐴𝐴𝑧𝑧 + 𝐵𝐵𝑧𝑧 𝑎𝑎
�𝑧𝑧
⇒Vector addition is commutative (𝐴𝐴⃗ + 𝐵𝐵=𝐵𝐵 + 𝐴𝐴),
⃗
⃗ 𝐶𝐶)=
distributive [𝐴𝐴(𝐵𝐵+ ⃗ 𝐴𝐴𝐵𝐵+
⃗ 𝐴𝐴⃗𝐶𝐶],
⃗ and
⃗
associative [𝐴𝐴+(𝐵𝐵+ ⃗ 𝐴𝐴+𝐵𝐵)+
𝐶𝐶)=( ⃗ ⃗
𝐶𝐶].

Fig. Addition of vectors (a) parallelogram rule, (b) head-to-tail rule[1]

6
 Vector subtraction
• Two vectors subtraction can be done as 𝐶𝐶= ⃗ 𝐴𝐴⃗ − 𝐵𝐵 = 𝐴𝐴⃗ +(−𝐵𝐵).
• The subtraction rule for two vectors is given below.
⃗ 𝑥𝑥 𝑎𝑎
Suppose 𝐴𝐴=𝐴𝐴 �𝑧𝑧 , and 𝐵𝐵=𝐵𝐵𝑥𝑥 𝑎𝑎
�𝑥𝑥 + 𝐴𝐴𝑦𝑦 𝑎𝑎�𝑦𝑦 + 𝐴𝐴𝑧𝑧 𝑎𝑎 �𝑧𝑧 , then,
�𝑥𝑥 + 𝐵𝐵𝑦𝑦 𝑎𝑎�𝑦𝑦 + 𝐵𝐵𝑧𝑧 𝑎𝑎
𝐷𝐷=𝐴𝐴⃗ − 𝐵𝐵 = 𝐴𝐴𝑥𝑥 − 𝐵𝐵𝑥𝑥 𝑎𝑎
�𝑥𝑥 + 𝐴𝐴𝑦𝑦 − 𝐵𝐵𝑦𝑦 𝑎𝑎�𝑦𝑦 + 𝐴𝐴𝑧𝑧 − 𝐵𝐵𝑧𝑧 𝑎𝑎
�𝑧𝑧
⇒Vector subtraction is distributive
[𝐴𝐴(𝐵𝐵 ⃗ 𝐴𝐴𝐵𝐵
⃗ − 𝐶𝐶)= ⃗ − 𝐴𝐴⃗𝐶𝐶].
⃗

Fig. Subtraction of vectors (a) parallelogram rule,

(b) head-to-tail rule[1]

7
 Examples-vector addition and subtraction
• If vector A and vector B are defined as:
𝐴𝐴⃗ = 5𝑖𝑖�𝑥𝑥 + 7𝑖𝑖�𝑦𝑦 + 𝑖𝑖�𝑧𝑧
𝐵𝐵 = 4𝑖𝑖�𝑥𝑥 + 6𝑖𝑖�𝑦𝑦 − 2𝑖𝑖�𝑧𝑧
• 𝐴𝐴⃗ + 𝐵𝐵 =?
• 𝐴𝐴⃗ − 𝐵𝐵 =?

8
 Examples-vector addition and subtraction
(cont.)
• Solution:
• 𝐴𝐴⃗ + 𝐵𝐵 = 9𝑖𝑖�𝑥𝑥 + 13𝑖𝑖�𝑦𝑦 − 𝑖𝑖�𝑧𝑧
• 𝐴𝐴⃗ − 𝐵𝐵 = 𝑖𝑖�𝑥𝑥 + 𝑖𝑖�𝑦𝑦 + 3𝑖𝑖�𝑧𝑧

9
 Position vector & distance vector
• The position vector of a point 𝑃𝑃 is directed from the
origin 𝑂𝑂 to 𝑃𝑃, and denoted by 𝑂𝑂𝑂𝑂.
• In the figure, the position vector
𝑂𝑂𝑂𝑂 = 𝒓𝒓𝒑𝒑 = 3�
𝑎𝑎𝑥𝑥 + 4�
𝑎𝑎𝑦𝑦 +5�
𝑎𝑎𝑧𝑧

Fig. Position vector[1]

• Distance vector is the displacement from one point
to other point.
• In the figure, 𝒓𝒓𝑷𝑷 and 𝒓𝒓𝑸𝑸 are the position vectors and
𝒓𝒓𝑷𝑷𝑷𝑷 is the distance vector, and can be calculated as:
𝒓𝒓𝑷𝑷𝑷𝑷 = 𝒓𝒓𝑸𝑸 -𝒓𝒓𝑷𝑷
or 𝑃𝑃𝑃𝑃=𝑂𝑂𝑂𝑂-𝑂𝑂𝑂𝑂 Fig. Distance vector[1]
10
 Vector multiplication
⇒There are two types of vector multiplications:
1. Scalar multiplication [dot (.) product]
2. Vector multiplication [cross(×) product]

Dot product- The dot product of two vectors is defined as the multiplication of
their magnitudes and the cosine of the angle between the vectors.
⃗ 𝐵𝐵 = 𝐴𝐴⃗ 𝐵𝐵 cos𝜃𝜃, where 𝜃𝜃 is the angle between the vector 𝐴𝐴⃗ and 𝐵𝐵.
𝐴𝐴.
⇒If two vectors are orthogonal to each other, the dot product is zero.
=> The dot product follows the following rules:
- The dot product is commutative, i.e. 𝐴𝐴. ⃗ 𝐵𝐵 = 𝐵𝐵. 𝐴𝐴⃗
⃗ 𝐵𝐵 + 𝐶𝐶⃗ = 𝐴𝐴.
- The dot product is distributive, i.e. 𝐴𝐴. ⃗ 𝐵𝐵 + 𝐴𝐴.
⃗ 𝐶𝐶⃗

11
 Vector multiplication (cont.)
� 𝑘𝑘.
⇒𝚤𝚤.̂ 𝚥𝚥̂ = 𝚥𝚥.̂ 𝑘𝑘= � 𝚤𝚤=0;
̂ 𝚤𝚤.̂ 𝚤𝚤̂ = 𝚥𝚥.̂ 𝚥𝚥= � 𝑘𝑘=1
̂ 𝑘𝑘. �
⇒ If 𝐴𝐴⃗ = 𝐴𝐴1 𝚤𝚤̂ + 𝐴𝐴2 𝚥𝚥̂ + 𝐴𝐴3 𝑘𝑘, � and 𝐵𝐵 = 𝐵𝐵1 𝚤𝚤̂ + 𝐵𝐵2 𝚥𝚥̂ + 𝐵𝐵3 𝑘𝑘� , then
⃗ 𝐵𝐵=𝐴𝐴1 𝐵𝐵1 +𝐴𝐴2 𝐵𝐵2 +𝐴𝐴3 𝐵𝐵3 .
𝐴𝐴.
 
 ˆ ˆ ˆ 
A.B ⇒  A i + A j + A k . B i + B j + B k 
  ˆ ˆ ˆ 
 1 2 3  1 2 3 
=  A iˆ . B iˆ  +  A iˆ . B ˆj  +  A iˆ . B kˆ  +
 1  1   1  2   1  3 

 A ˆ
j 
.
 B ˆ
i  
+
  A ˆj 
. B ˆ
j  
+
  A ˆ
j 
 . B kˆ 
 +
 2  1   2  2   2  3 

 A kˆ . B iˆ  +  A kˆ . B ˆj  +  A kˆ . B kˆ 

        
 3  1   3  2   3  3 
=AB +A B +A B
11 2 2 3 3
12
 Vector multiplication (cont.)
The cross product:
⇒The cross product is the multiplication of the magnitude of the vectors
and the sine of the angle between the vectors.
⇒The direction of the cross product is the direction of the thumb for right
handed screw as 𝐴𝐴⃗ is turned in to 𝐵𝐵.
⇒ 𝐴𝐴⃗ × 𝐵𝐵 = 𝐴𝐴⃗ 𝐵𝐵 sin𝜃𝜃�
𝑎𝑎𝑛𝑛 , where 𝑎𝑎�𝑛𝑛 represents the direction of the cross
product.

Fig. Direction of 𝐴𝐴⃗ × 𝐵𝐵 vector (a) right hand rule, 13

(b) right handed screw rule[1]
 Vector multiplication (cont.)
=> The cross product is useful in obtaining the unit vector normal to the plane containing
⃗
𝐴𝐴×𝐵𝐵
⃗
the two vectors 𝐴𝐴 and 𝐵𝐵: 𝑎𝑎�𝑛𝑛 = ⃗
𝐴𝐴 𝐵𝐵 sin𝜃𝜃
=> The cross product follows the following rules:
 The cross product is anti-commutative, i.e. 𝐴𝐴⃗ × 𝐵𝐵 = −𝐵𝐵 × 𝐴𝐴⃗
 The cross product is distributive, i.e. 𝐴𝐴⃗ × 𝐵𝐵 + 𝐶𝐶⃗ = 𝐴𝐴⃗ × 𝐵𝐵 + 𝐴𝐴⃗ × 𝐶𝐶⃗
 The cross product doesn’t follow the associative property (𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶⃗ ≠ 𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶⃗ )
� 𝚥𝚥̂ × 𝑘𝑘=
 𝚤𝚤̂ × 𝚥𝚥̂ = 𝑘𝑘, � 𝚤𝚤,̂ 𝑘𝑘� × 𝚤𝚤=
̂ 𝚥𝚥̂
� 𝑘𝑘� × 𝚥𝚥=-
 𝚥𝚥̂ × 𝚤𝚤̂ = −𝑘𝑘, � 𝚥𝚥̂
̂ 𝚤𝚤,̂ 𝚤𝚤̂ × 𝑘𝑘=−

Fig. Cross product using clock rotation (a) moving clockwise, (b)
moving anti-clockwise[1] 14
 Vector multiplication (cont.)
� 𝐵𝐵=𝐵𝐵1 𝚤𝚤+̂ 𝐵𝐵2 𝚥𝚥̂ + 𝐵𝐵3 𝑘𝑘�
⃗ 1 𝚤𝚤+̂ 𝐴𝐴2 𝚥𝚥̂ + 𝐴𝐴3 𝑘𝑘,
⇒If 𝐴𝐴=𝐴𝐴
 
A × B ⇒  A iˆ + A ˆj + A kˆ  ×  B iˆ + B ˆj + B kˆ 
 1 2 3   1 2 3 
( )( )( )( )( )
= A iˆ × B iˆ + A iˆ × B ˆj + A iˆ ×  B kˆ  +
1 1 1 2 1  3 
( )( )( )( )( )
A ˆj × B iˆ + A ˆj × B ˆj + A ˆj ×  B kˆ  +
2 1 2 2 2  3 
( )
 3 
( )
 A kˆ  × B iˆ +  A kˆ  × B ˆj +  A kˆ  ×  B kˆ 
1  3  2  3   3 
= 0 + A B kˆ − A B ˆj − A B kˆ + 0 + A B iˆ + A B ˆj − A B iˆ + 0
1 2 1 3 2 1 2 3 3 1 3 2
( ) ( ) (
= A B − A B iˆ − A B − A B ˆj + A B − A B kˆ
2 3 3 2 1 3 3 1
)
1 2 2 1

𝚤𝚤̂ 𝚥𝚥̂ 𝑘𝑘�

=> 𝐴𝐴⃗ × 𝐵𝐵= 𝐴𝐴1 𝐴𝐴2 𝐴𝐴3
𝐵𝐵1 𝐵𝐵2 𝐵𝐵3
15
 Example-scalar and vector multiplication
     
Q. Let : A = 2 xˆ + 4 yˆ ; B = xˆ + 7 yˆ . Solve A.B, A × B.

16
 Example-scalar and vector multiplication
(cont.)
• Sol.  
A.B = (2 xˆ + 4 yˆ )( . xˆ + 7 yˆ ) = 2 + 28 = 30
 
A × B = (2 xˆ + 4 yˆ )× ( xˆ + 7 yˆ )
= (2 xˆ )× ( xˆ ) + (2 xˆ )× (7 yˆ ) + (4 yˆ )× ( xˆ ) + (4 yˆ )× (7 yˆ )
= 0 + 14 zˆ − 4 zˆ + 0 = 10 zˆ
 
zˆ is ⊥ to A, B
     
ˆ
A × B = A B sin θiN = A B sin θzˆ
 
A = (2 ) + (4 ) = 4.472; B = (1) + (7 ) = 7.071
2 2 2 2

 
A.B = (2 xˆ + 4 yˆ )(
. xˆ + 7 yˆ ) = 2 + 28 = 30
 
(
   
)
A B = 31.62; cos θ = A.B A B = 0.9487 ⇒ θ = 18.43o
 
∴ A B sin θ = 31.62 x0.3161 = 9.996 ≅ 10
17
 Scalar triple product
⇒The scalar product of three vectors (scalar triple product) 𝐴𝐴, ⃗ 𝐵𝐵 and 𝐶𝐶⃗ is defined as
⃗ 𝐵𝐵 × 𝐶𝐶⃗ .
𝐴𝐴.
⃗ 1 𝚤𝚤+
⇒If 𝐴𝐴=𝐴𝐴 � 𝐵𝐵=𝐵𝐵1 𝚤𝚤+̂ 𝐵𝐵2 𝚥𝚥̂ + 𝐵𝐵3 𝑘𝑘,
̂ 𝐴𝐴2 𝚥𝚥̂ + 𝐴𝐴3 𝑘𝑘, � and 𝐶𝐶=𝐶𝐶 ̂ 𝐶𝐶2 𝚥𝚥̂ + 𝐶𝐶3 𝑘𝑘�
⃗ 1 𝚤𝚤+
iˆ1 iˆ2 iˆ3
  
A.BxC = ( A1iˆ1 + A2iˆ2 + A3iˆ3 ). B1 B2 B3
C1 C2 C3
A1 A2 A3
  
⇒ A.BxC = B1 B2 B3
C1 C2 C3

⇒Since the value of the determinant on the right side remains unchanged if the rows are
⃗ 𝐵𝐵 × 𝐶𝐶⃗ = 𝐶𝐶.
interchanged in a cylindrical manner, we have 𝐴𝐴. ⃗ 𝐴𝐴⃗ × 𝐵𝐵 = 𝐵𝐵. 𝐶𝐶⃗ × 𝐴𝐴⃗
18
 Vector triple product
⃗ 𝐵𝐵 and 𝐶𝐶⃗ is defined as 𝐴𝐴⃗ ×
⇒The vector product of three vectors (vector triple product) 𝐴𝐴,
𝐵𝐵 × 𝐶𝐶⃗ .
⇒𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶⃗ =𝐵𝐵 𝐴𝐴.
⃗ 𝐶𝐶⃗ − 𝐶𝐶⃗ 𝐴𝐴.
⃗ 𝐵𝐵
⇒It should be noted that
𝐴𝐴⃗ 𝐵𝐵. 𝐶𝐶⃗ ≠ 𝐴𝐴.
⃗ 𝐵𝐵 𝐶𝐶⃗
𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶⃗ ≠ 𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶⃗
• As an example, let us assume the three vectors to unit vectors as follows:
𝐴𝐴⃗ = 𝚤𝚤;̂ 𝐵𝐵 = 𝚤𝚤;̂ 𝐶𝐶⃗ = 𝚥𝚥̂
𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶⃗ =𝚤𝚤̂ × 𝚤𝚤̂ × 𝚥𝚥̂ =𝚤𝚤̂ × 𝑘𝑘� = −𝚥𝚥̂
𝐴𝐴⃗ × 𝐵𝐵 × 𝐶𝐶=
⃗ 𝚤𝚤̂ × 𝚤𝚤̂ × 𝚥𝚥=0
̂ × 𝚥𝚥̂ = 0

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 Components of a vector
⇒Using vector product, the projection (scalar or vector) of the vector
in a given direction can be determined.
⃗ 𝑎𝑎�𝐵𝐵 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝐴𝐴𝐴𝐴 = 𝐴𝐴.
⇒𝐴𝐴𝐵𝐵 = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝜃𝜃𝐴𝐴𝐴𝐴 =|𝐴𝐴| ⃗ 𝑎𝑎�𝐵𝐵 (scalar component)

⃗ 𝑎𝑎�𝐵𝐵 𝑎𝑎�𝐵𝐵 (vector component)

⇒𝐴𝐴𝐵𝐵 = 𝐴𝐴𝐵𝐵 𝑎𝑎�𝐵𝐵 = 𝐴𝐴.

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 References
[1] M.N. Sadiku: Elements of Electromagnetics, Oxford University Press,
ISBN 0-19-510368-8
[2] “Electromagnetics and Applications Assignment Help”,
(http://www.assignmenthelp.net/assignment_help/Electromagnetics-
And-Applications)

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