Antennas. Yi Huang

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Figure 1.6 Complex plane

      In this model, multiplication by −1 corresponds to a rotation of 180 degrees about the origin. Multiplication by j corresponds to a 90‐degree rotation anti‐clockwise, and the equation j² = −1 is interpreted as saying that if we apply two 90‐degree rotations about the origin, the net result is a single 180‐degree rotation. Note that a 90‐degree rotation clockwise also satisfies this interpretation.

      Another representation of a complex number Z is to use the amplitude and phase form:

      (1.4)

      where A is the amplitude and φ is the phase of the complex number Z, which are also shown in Figure 1.6. The two different representations are linked by the following equations:

(1.5)

      1.3.2 Vectors and Vector Operation

A scalar is a one‐dimensional quantity that has magnitude only, whereas a complex number is a two‐dimensional quantity. A vector can be viewed as a three‐dimensional (3D) quantity, and a special one – it has both a magnitude and a direction. For example, force and velocity are vectors. A position in space is a 3D quantity, but it does not have a direction, thus it is not a vector. Figure 1.7 is an illustration of vector A in Cartesian coordinates. It has three orthogonal components (A_x, A_y, A_z) along the x, y, and z directions, respectively. To distinguish vectors from scalars, the letter representing the vector is printed in bold, as A or a, and a unit vector is printed in bold with a hat over the letter as or .

Figure 1.7 Vector A in Cartesian coordinates

      The magnitude of vector A is given by

      (1.6)

      Now let us consider two vectors A and B:

The addition and subtraction of vectors can be expressed as

      (1.7)

      Obviously, the addition obeys the commutative law, that is A + B = B + A.

Figure 1.8 shows what the addition and subtraction mean geometrically. A vector may be multiplied or divided by a scalar. The magnitude changes but its direction remains the same. However, the multiplication of two vectors is complicated. There are two types of multiplication: dot product and cross product.

Figure 1.8 Vector addition and subtraction

      The dot product of two vectors is defined as

      (1.8)

      where θ is the angle between vector A and vector B and cos θ is also called the direction cosine. The dot • between A and B indicates the dot product that results in a scalar, thus it is also called a scalar product. If the angle θ is zero, A and B are in parallel – the dot product maximized, whereas for an angle of 90 degrees, i.e. when A and B are orthogonal, the dot product is zero.

      It is worth noting that the dot product obeys the commutative law, that is, A • B = B • A.

      The cross product of two vectors is defined as

(1.9)

where is a unit vector normal to the plane containing A and B. The cross × between A and B indicates the cross product that results in a vector C, thus, it is also called a vector product. The vector C is orthogonal to both A and B, and the direction of C follows a so‐called right‐hand rule as shown in Figure 1.9. If the angle θ is zero or 180 degrees, that is, A and B are in parallel, the cross product is zero, whereas for an angle of 90 degrees, i.e. A and B are orthogonal, the cross product of these two vectors reaches a maximum. Unlike the dot product, the cross product does not obey the commutative law.

Figure 1.9 The cross product of vectors A and B

      The cross product may be expressed in determinant form as follows, which is the same as Equation (1.9) but it may be easier for some people to memorize:

      (1.10)

      Another important thing about vectors is that any vector can be decomposed into three orthogonal components (such as x, y, and z components) in 3D or two orthogonal components in a 2D plane.

      Example 1.1 Vector operation

      Vectors and . Find:

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