# Vector Algebra

• Chapter 1: Vector Algebra . %1.1Introduction . -Definition: Electromagnetics (EM): The study of the interaction between electric charges at rest and in motion. It entails the analysis, synthesis, physical interpretation and application of electric and magnetic fields. In other words, electromagnetics (EM) is a branch of physics or electrical engineering in which electric and magnetic phenomena are studied. -Examples of applications: 1.Microwaves 2.Antennas 3.Electric machines 4.Satellite communications 5.Bioelectromagnetics 6.Plasmas 7.Nuclear research 8.Fiber optics 9.Electromagnetics interference and compatibility 10.Electromechanical energy conversion 11.Radar meteorology 12.Remote sensing Shortwaves . -In physical medicine: Electromagnetics power Microwaves . -Examples of devices: 1.Transformers 2.Electric relays 3.Radio/TV 4.Telephone 5.Electric motors 6.Transmission lines 7.Waveguides 8.Antennas 9.Optical fibers 10.Radars 11.Lasers . . . -.Summary of the subject of this book: Where, : The vector differential operator : The electric flux density : The magnetic flux density : The electric field intensity : The magnetic field intensity : The volume charge density : The current density . Maxwell’s equations:     . . . %1.3Scalar and Vectors . -Definitions: Scalar: A quantity that has only magnitude. Examples: Time, mass, distance, temperature, entropy, electric potential, population. . Vector: A quantity that has both magnitude and direction. Examples: velocity, force, displacement, electric field intensity. *NOTE: Another class of physical quantities is called Tensors, of which scalars and vector are special cases.   Field: A function that specifies a particular quantity everywhere in a region. Examples:  - Scalar fields: Temperature distribution, sound intensity, electric potential, refractive index of a stratified medium. – Vector Fields: Gravitational force, velocity. . . . %1.4Unit Vectors . -Notes about notations: A vector as said above, has both magnitude and direction. The magnitude is written as or The vector it self as .Now, a unit vector along is defined as a vector whose magnitude is unity and its direction is along . That is,     . and thus,   . . . .  The vector in Cartesian (also called rectangular) coordinates may be represented as:      . .  And the magnitude of is found as follows:    .  and   . . . . . %1.5Victor Addition and Subtraction . . . . . . . -Where k and are scalars . . . . . . . . . %1.6Victor Addition and Subtraction . -Definitions: .Position vector (radius vector): The position vector of a point P is defined as the directed distance from the origin O to to P. That is,     . Distance vector (separation vector): The…