alt="Schematic illustration of complex plane"/>
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:
1.3.2 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 (Ax, Ay, Az) 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
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
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
