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

      (1.6c)

      In a cylindrical coordinate system, unit vectors are defined as

      (1.7)

      Hence

      (1.8a)

      (1.8b)

      (1.8c)

      In a spherical coordinate system, unit vectors are defined as

      (1.9)

      which leads to

      (1.10b)

      (1.10b)

      (1.10c)

      1.2.1.2 Vector Operations and Properties

       Dot Product

The dot product between two vectors results in a scalar quantity. One simple application example for a dot product is the work that is done by a force for displacement from one point to another. Consider vectors

and

shown in Figure 1.2a. The dot product of these two vectors are then expressed as

      (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.

       Cross Product

      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)

       Vector Operation Properties for Dot and Cross Products

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

      (1.14a)