vectors pdf - Dr. kaydar M Quboa

ENGINEERING

ELECTROMAGNETICS

ENGINEERING

ELECTROMAGNETICS

Dr. Kaydar M. Quboa

Asst. Prof.

Dept. of Electrical Engineering

College of Engineering

University of Mosul

2015-2016

Dr. Kaydar M. Quboa

Asst. Prof.

Dept. of Electrical Engineering

College of Engineering

University of Mosul

2015-2016

Text book

1-VECTOR ANALYSIS

1.1 Scalars And Vectors

1-VECTOR ANALYSIS

1.1 Scalars And Vectors

•

The term scalar refers to a quantity whose value may

be represented by a single (positive or negative) real

number. The x, y, and z we use in basic algebra are

scalars. Other scalar quantities are mass, density,

pressure and others.

•

A vector quantity has both a magnitude and a

direction in space. Force, velocity, acceleration, and a

straight line from the positive to the negative

terminal of a storage battery are examples of

vectors.

1.2 Vector Algebra

1.2 Vector Algebra
•

The addition of vectors follows the

parallelogram law shown in Figure 1.1 for the

sum of two vectors, A and B.

•

The rule for the subtraction of vectors follows from that

for addition, for we may always express A−B as A+(−B);

the sign, or direction, of the second vector is reversed,

and this vector is then added to the first by the rule for

vector addition.

•

Vectors may be multiplied by scalars. The magnitude of

the vector changes, but its direction does not when the

scalar is positive, although it reverses direction when

multiplied by a negative scalar.

•

Division of a vector by a scalar is a multiplication by the

reciprocal ( )ﻣﻘﻠوبof that scalar.

•

The multiplication of a vector by a vector is discussed

in Sections 1.6 and 1.7.

1.3 The Rectangular Coordinate System

•

In the rectangular coordinate system we set up three

coordinate axes mutually at right angles to each other

and call them the x, y, and z axes as shown in Figure 1.2.

Figure 1.2

1.4 Vector Components And Unit Vectors

•

To describe a vector in the rectangular coordinate

system, let us first consider a vector r extending

outward from the origin. A logical way to identify this

vector is by giving the three component vectors, lying

along the three coordinate axes, whose vector sum must

be the given vector. If the component vectors of the

vector r are x, y, and z, then r = x+y+z. The component

vectors are shown in Figure 1.3a.

Figure 1.3a

We use the symbol a for a unit vector and identify its

direction by an appropriate subscript. Thus ax , ay ,and az

are the unit vectors in the rectangular coordinate

system.They are directed along the x, y, and z axes,

respectively, as shown in Figure 1.3b.

Figure 1.3b

•

A vector rp pointing from the origin to point P(1, 2, 3) is written r

+ 2a

+ 3a

•

The vector from P to Q may be obtained by applying the rule of

vector addition. This rule shows that the vector from the origin to P

plus the vector from P to Q is equal to the vector from the origin to

Q. The desired vector from P(1, 2, 3) to Q(2,−2, 1) is:

•

= r

− r

= (2 − 1)a

+ (−2 − 2)a

+ (1 − 3)a

= a

− 4a

− 2a

•

The vectors r

, r

, and R

are shown in Figure 1.3c.

Figure 1.3c

•

Any vector B then may be described by:

•

B = Bx a

+ By a

+ Bz a

•

The magnitude of B written |B| or simply B, is

given by:

•

A unit vector in a given direction is a vector in

that direction divided by its magnitude.

•

A unit vector in the direction of the vector B is:

EXAMPLE: Specify the unit vector extending from the

origin toward the point G(2,−2,−1).

1.5 Vector Addition And Subtraction

1.5 Vector Addition and Subtraction

1.6 VECTOR MULTIPLICATION

•

Given two vectors A and B, the dot product, or

scalar product, is defined as:

The Dot Product

•

Let A = Axax + Ayay + Azaz and B = Bxax + Byay + Bzaz.

Because the angle between two different unit vectors of the

rectangular coordinate system is 90◦, we then have

ax · ay = ay · ax = ax · az = az · ax = ay · az = az · ay = 0

The remaining three terms involve the dot product of a unit

vector with itself, which is unity, giving finally:

A vector dotted with itself yields the magnitude squared, or

The Cross Product

•

The cross product A × B is a vector.

•

The magnitude of A × B is equal to the product

of the magnitudes of A, B, and the sine of the

smaller angle between A and B;

•

The direction of A×B is perpendicular to the

plane containing A and B and is along one of the

two possible perpendiculars which is in the

direction of advance of a right-handed screw as

A is turned into B. This direction is illustrated in

Figure 1.5.

Figure 1.5

•

As an equation we can write:

Note: ax×ay=az; ay×az=ax and az×ax=ay
then:

•

then,

•

or written as:

•

Example: if A = 2ax − 3ay + az and B = −4ax −

2ay + 5az, find A X B.
Solution:

1.7 Review of Rectangular Coordinate System

1.8 Cylindrical Coordinate System

1.9 The Spherical Coordinate System

Differential Volume