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Chapter 6: Problem 93

Explain how to plot a complex number in the complex plane. Provide an example with your explanation.

Short Answer

Expert verified
To plot a complex number in the complex plane, first identify the real and imaginary components from it. The real part of the complex number is the horizontal coordinate, and the imaginary part is the vertical coordinate. In the example \(z = 3 + 4i\), the point (3,4) in the complex plane represents the complex number.

Step by step solution

01

Identifying the Real and Imaginary Components

Given a complex number, the first step is identifying the real and imaginary parts. If a complex number is written as \(z = a + bi\), then 'a' is the real part and 'b' is the imaginary part. For example, if \(z = 3 + 4i\), the real part is 3 and the imaginary part is 4.
02

Plotting the Real Component

The real component of the complex number corresponds to the x-coordinate in the plane. From the complex number \(z = 3 + 4i\), the real component 3 will be plotted along the x-axis. It is placed 3 units to the right of the origin.
03

Plotting the Imaginary Component

The imaginary component corresponds to the y-coordinate in the plane. From \(z = 3 + 4i\), the imaginary component 4 will be plotted along the y-axis. Moving 4 units upwards from the previous point marked on the x-axis, the point in the plane representing the complex number is identified.
04

Marking the Point

The important step is to plot the number. In our example \(z = 3 + 4i\), this is the point (3,4) on the complex plane. This point is 3 units along the real axis and 4 units up the imaginary axis. Mark this point in the complex plane.

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Most popular questions from this chapter

What are parallel vectors?

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$4 \mathbf{u} \cdot(5 \mathbf{v}-3 \mathbf{w})$$

Solve the equation \(2 x^{3}+5 x^{2}-4 x-3=0\) given that -3 is a zero of \(f(x)=2 x^{3}+5 x^{2}-4 x-3\) (Section \(2.4,\) Example 6 )

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{j}$$
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