Chapter 2:

Vector Algebra

Electromagnetics for Engineers

F.T. Ulaby

2-1 Basic Laws of Vector Algebra

2-2 Orthogonal Coordinate Systems

2-3 Transformations between Coordinate Systems

2-1 Basic Laws of Vector Algebra

Magnitude and direction of a vector

Unit and base vectors

Equal and identical vectors

Addition and subtraction of vectors

Position and distance vectors

Vector multiplication: Simple product = scalarvector kA,

Scalar or dot product AB, Vector or cross product A x B (right-hand rule)

Scalar A(BxC), and vector triple products Ax(BxC).

Magnitude and direction of a vector

Unit and base vectors

Equal and identical vectors

Addition and subtraction of vectors

Position vectors

Distance vector

Vector multiplication

Simple product = scalarvector

Scalar or dot product, AB

Properties of the vectors dot product

Finding the smaller angle between two specified

vectors:

Vector or cross product, A x B

Properties of the vectors cross product

Scalar and vector triple products

Scalar triple product,

Vector triple product,

2-2 Orthogonal Coordinate Systems

(three perpendicular dimensional coordinates)

1. Cartesian Coordinates

Differential length, area, and volume in Cartesian coordinates

(cont) Cartesian Coordinates

Differential length:

Differential surface areas:

Differential volume:

2. Cylindrical Coordinates

(r, , z)

In cylindrical coordinates a vector is expressed as

Point P(r1,1, z1) in cylindrical coordinates:

(cont) Cylindrical Coordinates

Differential length:

Differential surface areas:

Differential volume:

3. Spherical Coordinates (R, , )

In spherical coordinates a vector is

expressed as

Point P(R1, 1, 1) in spherical coordinates:

(cont) Spherical Coordinates

Differential length:

Differential surface areas:

Differential volume:

2-3 Transformations between Coordinate Systems

A. Cartesian (x, y, z) to Cylindrical (r, , z)Transformations

Both systems share the coordinate z, and the relations between

the other two pairs of coordinates are:

The inverse relations are:

(cont) Cartesian to Cylindrical Transformations

Unit vectors relations:

The relations in terms of x and y:

,

The inverse relations are:

,

(cont) Cartesian to Cylindrical Transformations

Example: A vector in Cartesian coordinates can be

transformed into in cylindrical

coordinates as follows:

and conversely:

B. Cartesian (x, y, z) to Spherical (R, , )Transformations

The inverse relations are:

(cont) Cartesian to Spherical Transformations

The expressions of ( ) in terms of ( ) are

The inverse relations are:

(cont) Transformations between Coordinate Systems

Distance between Two Points in

all Three Orthogonal Coordinates

The distance d between points P1 and P2 is given by

In Cartesian coordinates

In cylindrical coordinates

In spherical coordinates