Vector Analysis

Vectors and Scalars
Knowledge of vectors and scalars is important when analysing electromagnetic fields. A vector is a quantity that has both magnitude and direction. Vectors are represented by boldface roman-type symbols (A). An arrow on the top of the letter often represents vector ( A ?). The magnitude of the vector is represented by |A| or simply A. Displacement, velocity, force and acceleration are examples of vectors. Different vectors with directions are shown in Fig.1.1 .A vector field is a function that specifies a vector quantity everywhere in a region. Examples are gravitational force on a body in space and the displacement of a plane in space. A scalar is a quantity with magnitude but no direction. Length, mass, time , temperature and any real number are examples of scalar quantities. A scalar field is a function that specifies a scalar quantity everywhere in a region. Examples are temperature distribution and electric potential in a room.

A
A
B - B

Fig.( 1- 1) : Vectors with directions
1.2 Vector Components
A vector can be resolved into two components, namely the horizontal component and the vertical component. The addition of these two components is equal to the original vector. In Fig.(1- 2) , a vector F is working at an angle of ? with the x-axis. The x-axis component of this vector is
Fx =F cos ? (1 – 1 )
They-axis component is
Fy =F sin ? (1 – 2 )


y
Fy F
? x
FX
Fig.( 1- 2) : Vectors with directions
Vectors F 1 and F2 are working at angles of ? 1 and ? 2 with the x - axis, respectively, which are shown in Fig.2.3. Here, the x -axis and y - axis components are

Fx1=F1 cos?1 (1- 3)
Fx2=F2 cos?2 (1- 4)
Fy1=F1 sin?1 (1- 5)
Fy2=F2 sin?2 (1- 6)
The sum of the horizontal components is
Fx =Fx1+Fx2 (1- 7)
Substituting Eqs. (1.3) and (1.4) into Eq. (1.7) yields
Fx =F1 cos ?1+F2 cos ?2 (1- 8)
y
F2
Fy2 F1
Fy1

X
FX1

FX2
Fig. (1- 3) : Two vectors with directions


The sum of the vertical components is

Fy =Fy1+Fy2 (1 - 9)

Substituting Eqs. (1.5) and (1.6) into Eq. (1.9) yields

Fy =F1sin?1+F2sin?2 (1 - 10)

Finally, the resultant vector can be determined as

Fr = ?(F_X^2+ F_y^2 ) (1 - 11)

1.3 Unit Vectors
A unit vector is a vector whose magnitude is 1. Unit vectors in three directions are
a x, a y and a z as shown in Fig.1.4 . The magnitudes of three unit vectors are


a x =(1,0, 0) (1-12)
a y =(0,1, 0) (1-13)
a z =(0,0, 1) (1-14)

A general representation of a vector A is shown in Fig. 1.5. The unit vector a A is
working in the same direction as the vector A. The unit vector can be expressed as


a_A= A/|A| =1 ( 1- 15 )