Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Represent the complex number geometrically. $$ (-3 i)(2-i) $$

Step-by-Step Solution

Answer

The complex number \((-3-6i)\) is represented by the point \((-3, -6)\) on the Argand plane.

Step 1: Calculate the Product

To find the product of the complex numbers \(-3i\) and \(2-i\), let us first distribute each term:\[(-3i)(2-i) = (-3i)(2) + (-3i)(-i)\]Calculate each part separately:1. \(-3i \times 2 = -6i\)2. \(-3i \times (-i) = 3i^2\)Recall that \(i^2 = -1\), so:\[3i^2 = 3(-1) = -3\]

Step 2: Combine the Parts

After calculating each part in the last step, we combine them:\[(-6i) + (-3) = -3 - 6i\]The result is the complex number \(-3 - 6i\).

Step 3: Plot the Complex Number on the Argand Plane

Plot the complex number \(-3 - 6i\) on the Argand plane. The complex number can be represented as the point \((-3, -6)\).1. On the Argand plane, the x-axis represents the real part \(-3\).2. The y-axis represents the imaginary part \(-6\).Plot the point where these values intersect at \((-3, -6)\).

Step 4: Illustrate the vector

From the origin \((0,0)\), draw a vector to the point \((-3,-6)\) on the Argand plane. This vector represents the complex number \(-3 - 6i\). The horizontal component is \(-3\) and the vertical component is \(-6\).

Key Concepts

Argand plane

The Argand plane is a visual tool for representing complex numbers. It consists of two perpendicular axes, much like the Cartesian coordinate system used in algebra. On the Argand plane, the horizontal axis represents the real part of a complex number. The vertical axis is used for the imaginary part.

For example, let's consider the complex number \(-3 - 6i\). Here, the real part is \(-3\), and the imaginary part is \(-6\). Plotting this on the Argand plane involves marking the point \((-3, -6)\):

• Move \(-3\) units along the horizontal (real) axis.
• Move \(-6\) units along the vertical (imaginary) axis.

These coordinates uniquely represent the complex number on the Argand plane. This representation allows us to understand the number in spatial terms, which is handy for visualization.

imaginary unit

The imaginary unit, denoted as \(i\), is a fundamental concept in dealing with complex numbers. It is defined by its property that \(i^2 = -1\). This definition allows complex numbers to extend the real number system, enabling solutions to equations that do not have solutions within real numbers alone.

Complex numbers are of the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In the given exercise, the step \(-3i imes (-i) = 3i^2\) uses the property of the imaginary unit:

• Recognize \(i^2 = -1\).
• Calculate \(3(-1) = -3\).

Understanding \(i\) is essential, as it is the building block for complex number calculations. It allows us to perform operations extending beyond the real number line.

vector representation

Complex numbers also have a powerful connection with vectors, especially when represented on the Argand plane. Each complex number corresponds to a vector originating from the origin, \((0,0)\), to a point \((a, b)\) denoting the complex number \(a + bi\).

For \(-3 - 6i\), this vector extends from \((0,0)\) to \((-3,-6)\). Consider the components of this vector:

• The horizontal component is \(-3\), representing the real part.
• The vertical component is \(-6\), representing the imaginary part.

Vector representation from the origin provides insight into both the magnitude and direction of the complex number, offering a geometric perspective that complements algebraic calculations.

geometric representation

Geometric representation of complex numbers enhances our understanding of their properties and behaviors. By mapping them to points or vectors on the Argand plane, we obtain a visual model. This model is not only helpful for plotting but also for performing operations like addition, subtraction, and finding magnitudes.

When you plot a complex number like \(-3 - 6i\), you effectively create a geometric illustration of its position in space:

• The horizontal distance from the origin represents the real part.
• The vertical distance approximates the imaginary part.
• The length of the line (or vector) from the origin to the point is the magnitude of the complex number.

This geometrical view aids in grasping concepts like the argument and polar form, revealing deeper insights into the behavior of complex numbers in mathematical operations.