Problem 2 - Complex Analysis for Mathematics and Engineering

Question

Locate \(z_{1}\) and \(z_{2}\) vectorially and use vectors to find \(z_{1}+z_{2}\) and \(z_{1}-z_{2}\) when (a) \(z_{1}=2+3 i\) and \(z_{2}=4+i\) (b) \(z_{1}=-1+2 i\) and \(z_{2}=-2+3 i\) (c) \(z_{1}=1+i \sqrt{3}\) and \(z_{2}=-1+i \sqrt{3}\)

Step-by-Step Solution

Answer

For (a), \(z_1+z_2=6+4i\), \(z_1-z_2=-2+2i\). For (b), \(z_1+z_2=-3+5i\), \(z_1-z_2=1-i\). For (c), \(z_1+z_2=2i\sqrt{3}\), \(z_1-z_2=2\).

Step 1: Understand Vector Representation

Complex numbers can be represented as vectors in the complex plane, where the real part is the x-component and the imaginary part is the y-component. With this understanding, you can visualize the addition and subtraction of complex numbers as vector operations.

Step 2: Locate Vectors for (a)

For part (a), locate vectors for \(z_1 = 2 + 3i\) and \(z_2 = 4 + i\) in the complex plane. Draw \(z_1\) from the origin to the point (2, 3) and \(z_2\) from the origin to the point (4, 1).

Step 3: Calculate \(z_1 + z_2\) for (a)

Add the vectors by adding their corresponding components: \((2, 3) + (4, 1) = (6, 4)\). So, \(z_1 + z_2 = 6 + 4i\).

Step 4: Calculate \(z_1 - z_2\) for (a)

Subtract the vectors by subtracting their corresponding components: \((2, 3) - (4, 1) = (-2, 2)\). So, \(z_1 - z_2 = -2 + 2i\).

Step 5: Locate Vectors for (b)

For part (b), locate vectors for \(z_1 = -1 + 2i\) and \(z_2 = -2 + 3i\) in the complex plane. Draw \(z_1\) from the origin to the point (-1, 2) and \(z_2\) from the origin to the point (-2, 3).

Step 6: Calculate \(z_1 + z_2\) for (b)

Add the vectors by adding their corresponding components: \((-1, 2) + (-2, 3) = (-3, 5)\). So, \(z_1 + z_2 = -3 + 5i\).

Step 7: Calculate \(z_1 - z_2\) for (b)

Subtract the vectors by subtracting their corresponding components: \((-1, 2) - (-2, 3) = (1, -1)\). So, \(z_1 - z_2 = 1 - i\).

Step 8: Locate Vectors for (c)

For part (c), locate vectors for \(z_1 = 1 + i\sqrt{3}\) and \(z_2 = -1 + i\sqrt{3}\) in the complex plane. Draw \(z_1\) from the origin to the point (1, \(\sqrt{3}\)) and \(z_2\) from the origin to the point (-1, \(\sqrt{3}\)).

Step 9: Calculate \(z_1 + z_2\) for (c)

Add the vectors by adding their corresponding components: \((1, \sqrt{3}) + (-1, \sqrt{3}) = (0, 2\sqrt{3})\). So, \(z_1 + z_2 = 0 + 2i\sqrt{3}\).

Step 10: Calculate \(z_1 - z_2\) for (c)

Subtract the vectors by subtracting their corresponding components: \((1, \sqrt{3}) - (-1, \sqrt{3}) = (2, 0)\). So, \(z_1 - z_2 = 2\).

Key Concepts

Vector Representation

Complex numbers can sometimes seem abstract, but understanding them as vectors makes them much easier to handle intuitively. A complex number is composed of two parts: a real part and an imaginary part. By viewing these numbers as vectors, we essentially break them down into components:

• The real part of the complex number becomes the x-component of the vector.
• The imaginary part becomes the y-component of the vector.

To illustrate, consider the complex number \(z = a + bi\). This can be visualized as a vector \(\langle a, b \rangle\). The origin (0,0) of the complex plane serves as the starting point, and the endpoint is at \( (a, b) \). This graphical representation helps us easily perform operations like addition and subtraction as geometrical vector operations. These operations are simplified calculations of coordinates.

Complex Plane

The complex plane is a pivotal concept when it comes to working with complex numbers. It is a two-dimensional plane similar to the Cartesian coordinate system, but it is specialized for complex numbers. Here's a quick guide:

• The horizontal axis represents the real component of the complex number.
• The vertical axis represents the imaginary component.

Imagine plotting the number \(2 + 3i\). Move 2 units along the real axis (horizontal), then 3 units up along the imaginary axis (vertical). The resulting point (2, 3) helps visualize the complex number's position in space.

This plane not only aids in visualizing complex numbers but also turns operations into intuitive geometric manipulations. Plotting and analyzing these numbers relative to each other gives a compelling visual insight into their relationships and magnitudes.

Vector Operations

Understanding vector operations is crucial when dealing with complex numbers in the complex plane. Since we've established that complex numbers resemble vectors, we manipulate them using vector addition and subtraction.
To add two vectors \(\langle a, b \rangle\) and \(\langle c, d \rangle\), simply add the corresponding components:

• Real components: \(a + c\)
• Imaginary components: \(b + d\)
• Result: \(\langle a+c, b+d \rangle\)

Subtraction is just as straightforward. To subtract \(\langle c, d \rangle\) from \(\langle a, b \rangle\):

• Real components: \(a - c\)
• Imaginary components: \(b - d\)
• Result: \(\langle a-c, b-d \rangle\)

These operations mirror the algebraic operations on complex numbers \((a + bi) \pm (c + di)\). The vector addition/subtraction translates directly into combining/subtracting the real and imaginary parts separately, enabling straightforward computation.