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Chapter 6: Problem 24

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$-2+3 i$$

Short Answer

Expert verified
The complex number \( -2 + 3i \) is plotted by going two units left on the real axis and three units up on the imaginary axis. In polar form, it can be written as \( \sqrt{13}(cos 2.16 + i*sin 2.16)\).

Step by step solution

01

Plot the complex number

We can represent complex numbers graphically using the Cartesian coordinate system. In this system, the horizontal real axis corresponds to the real part of the complex number, and the vertical imaginary axis corresponds to the imaginary part of the complex number. So for the complex number \( -2 + 3i \), we go two units left on the real axis (since the real part is -2), and three units up on the imaginary axis (since the imaginary part is 3).
02

Convert the complex number to polar form

The conversion of a complex number from Cartesian form (a + bi) to polar form (r(cos 胃 + i*sin 胃)) involves first determining the modulus r and the argument 胃. The modulus r is calculated by \[r = \sqrt{a^2 + b^2}\]Substituting a = -2 and b = 3 gives \[r = \sqrt{(-2)^2 + (3)^2} = \sqrt{13}\]The argument 胃 can be found using the atan2 function in many programming languages or calculators: \[\胃 = atan2(b, a)\]Substituting a = -2 and b = 3 gives \[胃 = atan2(3, -2) 鈮� 2.158798930342464\] in radians.
03

Write the complex number in polar form

Expressing the complex number in polar form gives us \( -2 + 3i = \sqrt{13}(cos 2.16 + i*sin 2.16)\).

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Most popular questions from this chapter

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

Use a graphing utility to graph \(r=1+2 \sin n \theta\) for \(n=1,2,3,4,5,\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of large and small petals that occur corresponding to each value of \(n\) ? How are the large and small petals related when \(n\) is odd and when \(n\) is even?

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

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. $$4 \mathbf{u} \cdot(5 \mathbf{v}-3 \mathbf{w})$$

Will help you prepare for the material covered in the next section. Refer to Section 2.1 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i\). $$\frac{2+2 i}{1+i}$$
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