Abdullah Eroglu - RF/Microwave Engineering and Applications in Energy Systems

(1.11)

If θ between two vectors is 90°, then the dot product of these two vectors is equal to zero.

Figure 1.2 (a) Representation of vectors and for dot product. (b) Representation of vectors and for cross product.

Figure 1.3 Unit vector representation in a Cartesian coordinate system.

The cross product between two vectors is also a vector. The magnitude of the cross product of and is equal to the area of the parallelogram which is formed by these vectors, as shown in Figure 1.2b. Consider the two vectors given in Figure 1.2b. The cross product of these two vectors can be written as

(1.12)

is the unit vector along the normal to the plane containing A and B . It is important to note that

(1.13)

Some of the properties for dot and cross products are given below

(1.14a)

(1.14b)

(1.15a)

(1.15b)

(1.16a)

(1.16b)

(1.16c)

(1.16d)

Three coordinate systems are defined in this section: the rectangular or Cartesian coordinate system, the cylindrical coordinate system, and the spherical coordinate system.

Unit vector representation for a Cartesian coordinate system is shown in Figure 1.3.

A vector representation in a Cartesian coordinate system is given by (1.17).

(1.17)

The magnitude of vector is found from

(1.18)

The vector can be illustrated in the coordinate system, as shown in Figure 1.4. The base vector properties can be given by

(1.19)

(1.20)

Figure 1.4 Representation of vector in a Cartesian coordinate system.

Figure 1.5 Representation of vector in a cylindrical coordinate system.

The vector operations for dot and cross products for vectors and are given by

(1.21)