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Chapter 6: Problem 7

Plot each complex number and find its absolute value. $$z=3-i$$

Short Answer

Expert verified
The complex number \(z = 3 - i\) is represented on the complex plane by a point located 3 units to the right and 1 unit below the origin. It has an absolute value of \(\sqrt{10} = 3.1623\).

Step by step solution

01

Plotting The Complex Number

In the complex plane, the horizontal axis denotes the real part of the complex number, while the vertical axis represents the imaginary part. Here, the real part is 3, and the imaginary part is -1. Therefore, in order to plot the number \(z = 3 - i\) on the complex plane, start at the origin and move 3 units to the right along the real axis and then 1 unit downwards along the imaginary axis.
02

Calculating The Absolute Value

To find the absolute value (or modulus) of a complex number, the feeling is to apply the Pythagorean theorem. The absolute value of \(z = a + bi\) is given by \(\sqrt{a^2 + b^2}\). In this case, \(a=3\) and \(b=-1\), therefore, the absolute value of \(z = 3 - i\) is \(\sqrt{3^2 + (-1)^2} = \sqrt{10} = 3.1623\) when calculated to 4 decimal places.

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Most popular questions from this chapter

Exercises \(81-83\) will help you prepare for the material covered in the next section. Find the obtuse angle \(B,\) rounded to the nearest degree, satisfying $$ \cos B=\frac{6^{2}+4^{2}-9^{2}}{2 \cdot 6 \cdot 4} $$

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=3 \cos \frac{5 \pi}{3} \mathbf{i}+3 \sin \frac{5 \pi}{3} \mathbf{j}, \quad \mathbf{w}=2 \cos \pi \mathbf{i}+2 \sin \pi \mathbf{j}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There are no points on my graph of \(r^{2}=9 \cos 2 \theta\) for which \(\frac{\pi}{4}<\theta<\frac{3 \pi}{4}\)

Use the vectors $$\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad \text { and } \quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j},$$ to prove the given property. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Exercises \(81-83\) will help you prepare for the material covered in the next section. Simplify and round to the nearest whole number: $$ \sqrt{26(26-12)(26-16)(26-24)} $$
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