Problem 13 - Complex Analysis for Mathematics and Engineering

Question

Show that the equation of the line through the points \(z_{1}\) and \(z_{2}\) can be expressed in the form \(z=z_{1}+t\left(z_{2}-z_{1}\right)\), where \(t\) is a real number.

Step-by-Step Solution

Answer

The line through points \(z_1\) and \(z_2\) is expressed as \(z = z_1 + t(z_2 - z_1)\), where \(t\) is real.

Step 1: Understand the Concept of a Line in the Complex Plane

First, we need to understand that a line in the complex plane through two points can be thought of similarly to a line in the Cartesian coordinate plane. If you have two points \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\), the line can be expressed as the set of points \(z\) where \(z = z_1 + t(z_2 - z_1)\), with \(t\) being a real number.

Step 2: Express the Points as Complex Numbers

Consider the points \(z_1 = x_1 + iy_1\) and \(z_2 = x_2 + iy_2\) which are complex numbers. The difference between these two points is \(z_2 - z_1 = (x_2 - x_1) + i(y_2 - y_1)\), which gives us the direction of the line in the complex plane.

Step 3: Parametric Equation of the Line

The parametric equation for a line extending from \(z_1\) towards \(z_2\) can be written as \(z = z_1 + t(z_2 - z_1)\), where \(t\) is any real number. Here, \(t\) scales the difference \(z_2 - z_1\), moving the point \(z\) along the direction of the line.

Step 4: Substitute Difference into Parametric Equation

Substitute \(z_2 - z_1 = (x_2 - x_1) + i(y_2 - y_1)\) into the equation. This results in \(z = z_1 + t((x_2 - x_1) + i(y_2 - y_1))\), demonstrating that \(z\) is indeed any point on the line through \(z_1\) and \(z_2\).

Step 5: Verify the Line Behaves as Expected

When \(t = 0\), \(z = z_1\), and when \(t = 1\), \(z = z_2\). Thus, the equation correctly expresses the line passing through both \(z_1\) and \(z_2\). This verifies our parametric representation as accurate.

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane where complex numbers are represented as points. This is also known as the Argand plane. Complex numbers are of the form \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part. The complex plane helps visualize complex numbers and their operations, much like the Cartesian coordinate system does for real numbers.

• Real axis: This is the horizontal axis representing the real part of the complex number.
• Imaginary axis: This is the vertical axis representing the imaginary part of the complex number.

Complex numbers can be graphed as points or vectors from the origin to the point \((x, y)\). This visual representation allows us to utilize geometric concepts, such as lines, distances, and angles, in the complex number context.

Parametric Equations

Parametric equations are a way of defining a set of points using one or more parameters, typically describing a path or a curve in a space. When it comes to lines in the complex plane, a parametric equation describes the line using a parameter, usually denoted by \( t \). In the context of complex numbers, consider two points \( z_1 \) and \( z_2 \) in the complex plane. The line passing through these points can be expressed with the parametric equation:\[ z = z_1 + t(z_2 - z_1) \]

• \( z_1 + t(z_2 - z_1) \): \( z \) represents a point on the line. \( z_1 \) is a starting point, while \( z_2 - z_1 \) gives the direction of the line.
• \( t \): This real parameter 'stretches' or 'shrinks' the segment \( z_2 - z_1 \). By changing \( t \), we can locate any point along the line.

Through this parametric form, the entire line connecting \( z_1 \) and \( z_2 \) is captured. This method is powerful for describing geometric paths like lines, circles, and more.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part, typically written as \( z = x + iy \), where \( x \) is the real part and \( iy \) is the imaginary part. Each complex number corresponds to a point in the complex plane. Key properties and operations with complex numbers include:

• Conjugate: The complex conjugate of \( z = x + iy \) is \( \overline{z} = x - iy \). It reflects the point across the real axis.
• Magnitude: The magnitude (or modulus) of \( z \) is \( |z| = \sqrt{x^2 + y^2} \). This is the distance of the point from the origin.
• Addition/Subtraction: Two complex numbers are added by summing their real parts and imaginary parts separately.
• Multiplication: Multiply by expanding \( (x + iy)(a + ib) = (xa - yb) + i(xb + ya) \).

Understanding complex numbers is crucial in fields such as electrical engineering, quantum physics, and many branches of mathematics, as they allow for solutions and explanations that real numbers alone cannot provide.