# FWLec1 - A. F. Peterson: Notes on Electromagnetic Fields &...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 Fields & Waves Note #1 Vector Algebra Objectives: Review vectors, dot products, cross products, related calculations, and basic notation in Cartesian coordinates. In our study of electromagnetic fields, various quantities will be described in three- dimensional space. Our reference frame will be a right-handed Cartesian coordinate system (Figure 1). The system is deemed “right-handed” by virtue of the ordering (x-axis before y-axis before z-axis) and the fact that if the x-axis is rotated toward the y-axis according to the spiral of a right-handed screw, the screw moves in the z-direction. In this three-dimensional world, we will work with scalar and vector quantities. Scalar entities have a number or “size” associated with them, while vector quantities have a size and a direction. With vector quantities, the size is usually associated with the length or magnitude of the vector. For example, Ax = and Bx yz =+ - 34 ˆˆ ˆ are vector quantities. The first denotes a vector of length 3 oriented parallel to the x-axis. The second denotes a vector skewed with respect to the three principal axes, with a length of 3 along the x direction, 1 along the y direction, and 4 along the –z direction. The quantities 3, 1, and –4 are the components of the vector. Vector Addition There are a number of ways in which vector quantities can be manipulated. Simple operations such as addition or subtraction are well known, as illustrated by + ( + ) +( ) +( 32 36 33 11 26 64 ˆˆ xy z xy z xz ++ -- =- To add two vectors, one simply adds the coefficients of the various components. Subtraction works in a similar manner.

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 9/04 For convenience, we will denote general vector quantities by a letter with a bar over the top: Ax yz =+ + 32 ˆˆ ˆ Vectors that have a length equal to one are denoted using a carat or “hat” over the top: ˆ, ˆ, ˆ xyz These unit vectors are the principal axes of the coordinate system. Other unit vectors will be introduced below. Multiplication of scalars with vectors Besides addition and subtraction, vectors can be multiplied with scalars or with other vectors. Suppose + If we multiply this vector with the scalar 6, we obtain 66 36 16 2 18 6 12 xy z = + + + () ˆ ˆ ˆ ˆˆ ˆ

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## This note was uploaded on 01/27/2011 for the course ECE 3025 taught by Professor Citrin during the Spring '08 term at Georgia Tech.

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FWLec1 - A. F. Peterson: Notes on Electromagnetic Fields &...

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