Coordinate geometry is fundamental in Mechatronics Technology for describing positions, movements, and transformations of components in automation and robotics systems. Complex numbers provide a powerful way to represent two-dimensional coordinates and rotations, which are common in sensor data analysis, actuator control, and signal processing. Mastering complex geometry concepts such as modulus, argument, and conjugates enables precise modeling and computation in designing mechatronic systems.

3.1 Complex geometry

Complex geometry deals with numbers comprising a real part and an imaginary part, representing points in a plane. These complex numbers can be visualized on an Argand diagram, where the horizontal axis is the real part and the vertical axis is the imaginary part. Understanding the magnitude (modulus) and angle (argument) of complex numbers is essential for analyzing oscillations, phasors, and vector quantities in mechatronic circuits and controls.

3.1.1 Real part

The real part of a complex number represents its horizontal coordinate on the plane and corresponds to measurable quantities like displacement or voltage magnitude in mechatronics.

The general form of a complex number is$$z = x + yi$$ where \(x\) is the real part and \(y\) is the imaginary part.

The real part is denoted as \(\Re(z) = x\).

Worked Examples

Example 1: A sensor output is represented by the complex number \(z = 4 + 3i\). Find the real part.

Given: \(z = 4 + 3i\)

\[ \Re(z) = 4 \]

Answer: 4

Example 2: An actuator position is represented by \(z = -7 + 2i\). Determine the real part.

Given: \(z = -7 + 2i\)

\[ \Re(z) = -7 \]

Answer: -7

Example 3: In signal processing, the complex number is \(z = 0.5 - 1.2i\). Find the real part.

Given: \(z = 0.5 - 1.2i\)

\[ \Re(z) = 0.5 \]

Answer: 0.5

Example 4: A control system represents a state as \(z = -3.6 + 0i\). Find the real part.

Given: \(z = -3.6 + 0i\)

\[ \Re(z) = -3.6 \]

Answer: -3.6

3.1.2 Imaginary part

The imaginary part of a complex number corresponds to the vertical coordinate on the Argand diagram and often models phase or reactive components in mechatronic circuits.

For \(z = x + yi\), the imaginary part is \(\Im(z) = y\).

Worked Examples

Example 1: For \(z = 6 + 5i\), find the imaginary part.

Given: \(z = 6 + 5i\)

\[ \Im(z) = 5 \]

Answer: 5

Example 2: A complex voltage is \(z = -2 - 4i\). Determine the imaginary part.

Given: \(z = -2 - 4i\)

\[ \Im(z) = -4 \]

Answer: -4

Example 3: The state of a robotic joint is \(z = 3 + 0i\). Find the imaginary part.

Given: \(z = 3 + 0i\)

\[ \Im(z) = 0 \]

Answer: 0

Example 4: A signal is represented by \(z = 0.8 - 0.9i\). Find the imaginary part.

Given: \(z = 0.8 - 0.9i\)

\[ \Im(z) = -0.9 \]

Answer: -0.9

3.1.3 Argand diagram

The Argand diagram is a graphical representation of complex numbers with the horizontal axis as the real part and the vertical axis as the imaginary part. This diagram helps visualize complex numbers as vectors, useful in analyzing phasors and system responses.

A complex number \(z = x + yi\) is plotted at point \((x, y)\) on the Argand plane.

Worked Examples

Example 1: Plot the complex number \(z = 3 + 4i\).

Given: \(z = 3 + 4i\)

Coordinates: \((3, 4)\)

Plot the point 3 units right on the real axis and 4 units up on the imaginary axis.

Answer: Point at (3,4) on the Argand diagram.

Example 2: Plot \(z = -2 + 5i\).

Given: \(z = -2 + 5i\)

Coordinates: \((-2, 5)\)

Plot the point 2 units left and 5 units up.

Answer: Point at (-2,5).

Example 3: Plot \(z = -4 - 3i\).

Given: \(z = -4 - 3i\)

Coordinates: \((-4, -3)\)

Plot 4 units left and 3 units down.

Answer: Point at (-4,-3).

Example 4: Plot \(z = 0 + 6i\).

Given: \(z = 0 + 6i\)

Coordinates: \((0, 6)\)

Plot at origin horizontally and 6 units up.

Answer: Point at (0,6).

3.1.4 Modulus/magnitude

The modulus or magnitude of a complex number represents its distance from the origin on the Argand diagram, calculated using the Pythagorean theorem.

For \(z = x + yi\),$$|z| = \sqrt{x^2 + y^2}$$

The modulus indicates the amplitude in signals or the length of vectors in mechanical systems.

Worked Examples

Example 1: Find the modulus of \(z = 3 + 4i\).

Given: \(z = 3 + 4i\)

$$|z| = \sqrt{3^2 + 4^2}$$

$$= \sqrt{9 + 16}$$

$$= \sqrt{25}$$

Answer: 5

Example 2: Find the modulus of \(z = -5 + 12i\).

Given: \(z = -5 + 12i\)

$$|z| = \sqrt{(-5)^2 + 12^2}$$

$$= \sqrt{25 + 144}$$

$$= \sqrt{169}$$

Answer: 13

Example 3: Find the modulus of \(z = 7 - 24i\).

Given: \(z = 7 - 24i\)

$$|z| = \sqrt{7^2 + (-24)^2}$$

$$= \sqrt{49 + 576}$$

$$= \sqrt{625}$$

Answer: 25

Example 4: Find the modulus of \(z = -8 - 15i\).

Given: \(z = -8 - 15i\)

$$|z| = \sqrt{(-8)^2 + (-15)^2}$$

$$= \sqrt{64 + 225}$$

$$= \sqrt{289}$$

Answer: 17

3.1.5 Argument /angle

The argument of a complex number is the angle \(\theta\) it makes with the positive real axis in the Argand diagram, measured in radians or degrees. It is found using the inverse tangent of the ratio of the imaginary part to the real part.

For \(z = x + yi\),$$\theta = \tan^{-1}(\frac{y}{x})$$

The angle is crucial in representing phase shifts in signals and rotations in robotics.

Worked Examples

Example 1: Find the argument of \(z = 1 + \sqrt{3}i\) (in degrees).

Given: \(z = 1 + \sqrt{3}i\), \(x=1\), \(y=\sqrt{3}\)

$$\theta = \tan^{-1}(\frac{\sqrt{3}}{1})$$

$$= \tan^{-1}(\sqrt{3})$$

$$= 60^\circ$$

Answer: 60°

Example 2: Find the argument of \(z = -1 + \sqrt{3}i\) (in degrees).

Given: \(z = -1 + \sqrt{3}i\), \(x=-1\), \(y=\sqrt{3}\)

$$\theta = \tan^{-1}(\frac{\sqrt{3}}{-1})$$

$$= \tan^{-1}(-\sqrt{3})$$

Since \(x<0\) and \(y>0\), angle in 2nd quadrant:

$$\theta = 180^\circ - 60^\circ = 120^\circ$$

Answer: 120°

Example 3: Find the argument of \(z = -1 - \sqrt{3}i\) (in radians).

Given: \(z = -1 - \sqrt{3}i\), \(x=-1\), \(y=-\sqrt{3}\)

$$\theta = \tan^{-1}(\frac{-\sqrt{3}}{-1})$$

$$= \tan^{-1}(\sqrt{3})$$

Since \(x<0\), \(y<0\), angle in 3rd quadrant:

$$\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$$

Answer: \(\frac{4\pi}{3}\) radians

Example 4: Find the argument of \(z = 0 + 2i\).

Given: \(z = 0 + 2i\), \(x=0\), \(y=2\)

Angle lies on positive imaginary axis:

Answer: 90° or \(\frac{\pi}{2}\) radians

3.1.6 Conjugate

The conjugate of a complex number \(z = x + yi\) is \( \overline{z} = x, yi \). It reflects the point across the real axis on the Argand diagram. Conjugates are used to simplify division of complex numbers and solve equations in mechatronics control systems.

Worked Examples

Example 1: Find the conjugate of \(z = 4 + 3i\).

Given: \(z = 4 + 3i\)

\[ \overline{z} = 4 - 3i \]

Answer: 4 - 3i

Example 2: Find the conjugate of \(z = -2 + 5i\).

Given: \(z = -2 + 5i\)

\[ \overline{z} = -2 - 5i \]

Answer: -2 - 5i

Example 3: Find the conjugate of \(z = 0 - 7i\).

Given: \(z = 0 - 7i\)

\[ \overline{z} = 0 + 7i \]

Answer: 7i

Example 4: Find the conjugate of \(z = 3\).

Given: \(z = 3 + 0i\)

\[ \overline{z} = 3 - 0i = 3 \]

Answer: 3

Chapter Summary

This chapter introduced complex geometry by exploring the real and imaginary parts of complex numbers and their representation on the Argand diagram. It covered how to determine the modulus or magnitude and the argument or angle of a complex number, as well as the concept of the conjugate. The chapter then detailed the fundamental operations on complex numbers, including addition, subtraction, multiplication, and division, alongside methods for converting between polar and rectangular forms. De Moivre's theorem was examined for expanding powers of complex numbers and finding their roots. The chapter concluded with the application of complex numbers to derive trigonometric identities, linking algebraic and geometric interpretations. Throughout, the focus remained on practical techniques essential for solving coordinate geometry problems in engineering contexts.