Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Answer each of the following. The modulus of a complex number represents the __________ of the vector representing it in the complex plane.

Step-by-Step Solution

Answer

The length or magnitude.

Step 1: Understand the Complex Plane

To solve this question, first recognize that the complex plane is a two-dimensional plane where each point represents a complex number. A complex number has a real part and an imaginary part, plotted along the x-axis and y-axis, respectively.

Step 2: Define the Modulus of a Complex Number

The modulus of a complex number, denoted as \(|z|\) for a complex number \(z = a + bi\), measures the distance of the complex number from the origin in the complex plane. It is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\).

Step 3: Relate Modulus to Distance in Complex Plane

In the context of vectors, the modulus can also be thought of as the length or magnitude of the vector representing the complex number in the complex plane. This is because the modulus provides the measure of how far the point representing the complex number is from the origin.

Key Concepts

Modulus

The modulus of a complex number is a fundamental concept when dealing with complex numbers and their geometric representation. A complex number, generally expressed as \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, can be visualized as a point in the complex plane. The modulus of this complex number is essentially the measure of the distance from this point to the origin.To find the modulus, you can apply the formula \(|z| = \sqrt{a^2 + b^2}\). This formula is derived from the Pythagorean theorem, similar to how you would calculate the length of the hypotenuse of a right triangle.

• The real part \(a\) is your horizontal component.
• The imaginary part \(b\) is your vertical component.

Thus, the modulus gives you the length or magnitude of the vector formed with the point \((a, b)\) in the complex plane.

Complex Plane

The complex plane is an essential concept in the visualization of complex numbers. It is similar to the Cartesian coordinate system but specifically designed to represent complex numbers. In this plane:

• The horizontal axis, often called the real axis, represents the real part of the complex number.
• The vertical axis, also termed as the imaginary axis, represents the imaginary part.

Each complex number \(z = a + bi\) corresponds to a unique point \((a, b)\) in this plano. Thus, the complex plane allows us to represent and manipulate complex numbers geometrically, making it easier to comprehend operations like addition and multiplication. Importantly, it facilitates the understanding of concepts such as the modulus and argument of complex numbers.

Vectors in Complex Plane

In the complex plane, complex numbers can be treated as vectors. Visualizing complex numbers as vectors helps in interpreting operations and properties like addition, scaling, and particularly the concept of modulus.A vector in the complex plane originates from the origin and terminates at the point representing the complex number. The components of this vector are:

• The horizontal component \(a\), which aligns with the real axis.
• The vertical component \(b\), which aligns with the imaginary axis.

The modulus of the complex number is equivalent to the length of this vector. Thus, in vector terms, the modulus tells us the magnitude of the vector, showcasing its properties such as direction and scale. This conceptualization is critical in fields like physics and engineering, where complex numbers often represent forces, currents, or other vector quantities.