A.1 Vectors

This section gives a compact summary of the vector algebra for the linear three-dimensional vector space. For a more detailed treatment see for example [1, 2]. A vector is a variable quantity that has both magnitude and direction and can be resolved into components, such as force, electric, and magnetic field. In the Cartesian coordinate system ( x, y, z) a vector a is represented by its Cartesian coordinates ( a _x, a _y, a _z). The vector may be visualized by an arrow starting fom the origin and terminating at the point ( a _x, a _y, a _z), as depicted in Figure A.1(a). However, we could start from any point in our Cartesian reference frame. The origin only is chosen for simplicity.

Figure A.1: Cartesian components of a vector (a) in a right-handed coordinate system, and (b) in a left-handed coordinate system.

Let e _x be a vector of unit magnitude pointing in the positive x -direction and e _y and e _z vectors of unit magnitude pointing in the positive y- and z-directions, respectively. By vector addition we obtain

(A.1)

The vectors e _x, e _y, and e _z form a basis of our three-dimensional linear vector space. Figure A.1(a) shows the vector a in a right-handed Cartesian coordinate system ( x, y, z), whereas in Figure A.1(b) the...