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Chapter 6: Problem 1

Plot each complex number and find its absolute value. $$z=4 i$$

Short Answer

Expert verified
The absolute value of the complex number \( z=4i \) is 4.

Step by step solution

01

Understand the complex number

The given complex number is \( z=4i \), which consists only of an imaginary part. It is represented on the complex plane as a point on the positive imaginary axis at a distance of 4 units from the origin.
02

Plotting the complex number

On the complex plane, the x-axis represents the real part and the y-axis represents the imaginary part of a complex number. Since our complex number \( z=4i \) has no real part and has an imaginary part of 4, its graph is a point on the positive imaginary axis at a distance of 4 units above the origin.
03

Calculating the absolute value

The absolute value (or modulus) of a complex number is the distance from the origin to the point representing the complex number on the complex plane. Given a complex number \( z = re^{i胃} \), its absolute value is \( |z| = r \). In our case, our complex number \( z=4i \) can be rewritten as \( z = 0 + 4i = 0 + 4e^{i(蟺/2)} \) (since \( i = e^{i(蟺/2)} \)). Therefore, its absolute value is \( |z| = 4 \).

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Most popular questions from this chapter

The components of \(\mathbf{v}=240 \mathbf{i}+300 \mathbf{j}\) represent the respective number of gallons of regular and premium gas sold at a station. The components of \(\mathbf{w}=2.90 \mathbf{i}+3.07 \mathbf{j}\) represent the respective prices per gallon for each kind of gas. Find \(\mathbf{v} \cdot \mathbf{w}\) and describe what the answer means in practical terms.

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 \(\mathbf{w}.\) $$\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$$

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\cos ^{2} 5 \theta+\sin 3 \theta+0.3$$

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

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=3 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}+5 \mathbf{j}$$
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