Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Graph each complex number as a vector in the complex plane. Do not use a calculator. $$3 i$$

Step-by-Step Solution

Answer

The complex number \(3i\) is represented as a vertical vector at \((0, 3)\) on the complex plane.

Step 1: Understanding the Complex Number

The complex number given is \(3i\). In the form of \(a + bi\), the real part \(a = 0\) and the imaginary part \(b = 3\). This means it lies on the imaginary axis at \(b = 3\).

Step 2: Representation as a Vector

In the complex plane, a complex number \( a + bi \) is represented by the vector from the origin \((0,0)\) to the point \((a, b)\). For \(3i\), this point is \((0, 3)\).

Step 3: Graphing the Vector

To graph \(3i\), draw a vector that begins at the origin \((0, 0)\) pointing straight up to \((0, 3)\) on the imaginary axis. This represents the complex number \(3i\).

Key Concepts

Complex Plane

The complex plane is a two-dimensional plane used for visualizing complex numbers. It's quite similar to the Cartesian coordinate system you might be familiar with, but instead of just having an "x" and "y" axis, we call them the "real" and "imaginary" axes.
- The horizontal axis is the real axis where real numbers are located.- The vertical axis is the imaginary axis, which we'll discuss more later.
Each complex number corresponds to a unique point on this plane. If you have a complex number like \(a + bi\), \(a\) is the coordinate on the real axis, and \(b\) is the coordinate on the imaginary axis. It is mapped as the point \((a, b)\) on the complex plane. This makes the complex plane a powerful tool for visualizing and performing operations on complex numbers.

Imaginary Axis

The imaginary axis is the vertical component of the complex plane, akin to the "y" axis in the Cartesian coordinate system. It specifically hosts the imaginary components of complex numbers. Every point on the imaginary axis represents some multiple of the imaginary unit \(i\), which is defined as \(\sqrt{-1}\).
- A complex number purely on the imaginary axis has no real part, meaning it looks like \(0 + bi\).
In our example, the complex number given is \(3i\), which lies directly on the imaginary axis at the point \(b = 3\). This means that \(3i\) is simply a vertical movement of 3 units from the origin along this axis. Thus, every imaginary number can be visualized as a point above or below the origin on this axis.

Vector Representation

Complex numbers can be elegantly visualized as vectors in the complex plane. This representation helps in understanding how they relate to one another and are involved in operations like addition and multiplication.
- A vector can be thought of as something that has both a direction and a magnitude.
To represent a complex number such as \(3i\) as a vector, we start at the origin \((0, 0)\) and draw an arrow pointing to its corresponding point \((a, b)\). For \(3i\), this simply involves drawing a line from the origin straight upward to the point \((0, 3)\). This visualization as a vector effectively shows that \(3i\) has a direction along the imaginary axis and a magnitude of 3. Using vectors in this way gives us clearer insights into the structure and behavior of complex numbers in the plane.