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Chapter 9: Problem 66

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$(1-i)(3-3 i)$$

Short Answer

Expert verified
Question: Multiply the given complex numbers (1-i) and (3-3i), and express the result in rectangular form. Answer: 6

Step by step solution

01

Convert to Polar Form

To convert a complex number to polar form, we use the formulas \(r = \sqrt{a^2 + b^2}\) and \(\theta = \mathrm{atan2}(b, a)\), where \(a\) is the real part and \(b\) is the imaginary part. For the complex number \((1-i)\), we have: \(r = \sqrt{ (1)^2 + (-1)^2 } = \sqrt{2}\) \(\theta = \mathrm{atan2}(-1, 1) = -\frac{\pi}{4}\) So, \((1-i) = (\sqrt{2}, -\frac{\pi}{4})\) in polar form. For the complex number \((3-3i)\), we have: \(r = \sqrt{ (3)^2 + (-3)^2 } = 3\sqrt{2}\) \(\theta = \mathrm{atan2}(-3, 3) = -\frac{3\pi}{4}\) So, \((3-3i) = (3\sqrt{2}, -\frac{3\pi}{4})\) in polar form.
02

Multiply Polar Forms

To multiply two complex numbers in polar form, we multiply their magnitudes and add their angles: \(r = (\sqrt{2})(3\sqrt{2}) = 6\) \(\theta = -\frac{\pi}{4} + (-\frac{3\pi}{4}) = -\pi\) Hence, the product in polar form is \((6, -\pi)\).
03

Convert Result Back to Rectangular Form

To convert a complex number in polar form back to rectangular form, we use the formulas \(a = r\cos{\theta}\) and \(b = r\sin{\theta}\): \(a = 6\cos{(-\pi)} = 6\) \(b = 6\sin{(-\pi)} = 0\) So, the result in rectangular form is \(6 + 0i = 6\). So, \((1-i)(3-3i) = 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex numbers
Complex numbers are numbers that have both a real part and an imaginary part. The standard form to represent a complex number is \(a + bi\) where \(a\) is the real component and \(b\) is the imaginary component. The imaginary unit \(i\) is a special quantity defined as the square root of -1. When dealing with complex numbers, understanding their representation is crucial:
  • The real part \(a\) determines the position along the horizontal axis on the complex plane.
  • The imaginary part \(b\) corresponds to the position along the vertical axis.
Complex numbers can be operated on by addition, subtraction, multiplication, and division, similar to real numbers. However, their multiplication and division can be made simpler by using polar form, which provides a more intuitive approach to these operations.
polar coordinates
Polar coordinates are another way to represent complex numbers by specifying a magnitude and an angle, rather than the traditional Cartesian coordinates. This method is especially beneficial when performing multiplication or division of complex numbers, as the calculations become straightforward.Here鈥檚 how it works:
  • The magnitude \(r\) of a complex number is the distance from the origin to the point \(a + bi\) on the complex plane. It's calculated as \ \( r = \sqrt{a^2 + b^2} \) \.
  • The angle \(\theta\), also known as the argument, is the direction from the positive real axis to the line segment that represents the complex number. It can be found using the arctangent function, \ \( \theta = \mathrm{atan2}(b, a) \) \.
Representing complex numbers in polar form allows easier manipulation and understanding of their behavior when multiplied or divided.
rectangular form
Rectangular form, also known as the standard form, expresses a complex number as \(a + bi\). After calculations in polar form, results are often converted back to rectangular form for intuitive understanding.The conversion from polar form back to rectangular form involves trigonometric functions:
  • You calculate the real part \(a\) using: \ \( a = r\cos{\theta} \) \.
  • The imaginary part \(b\) is determined by: \ \( b = r\sin{\theta} \) \.
This conversion provides a clear visualization of the location of the complex number on the complex plane. Understanding both polar and rectangular forms allow for greater flexibility in mathematical calculations and visual interpretations.

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Most popular questions from this chapter

The sum of two distinct complex numbers, \(a+b i\) and \(c+d i,\) can be found geometrically by means of the socalled parallelogram rule: Plot the points \(a+b i\) and \(c+d i\) in the complex plane, and form the parallelogram, three of whose vertices are \(0, a+b i,\) and \(c+d i,\) as in the figure. Then the fourth vertex of the parallelogram is the point whose coordinate is the sum $$(a+b i)+(c+d i)=(a+c)+(b+d) i$$ (GRAPH CAN'T COPY). Complete the following proof of the parallelogram rule when \(a \neq 0\) and \(c \neq 0\) (a) Find the slope of the line \(K\) from 0 to \(a+b i .[\text { Hint: } K\) contains the points \((0,0) \text { and }(a, b) .]\) (b) Find the slope of the line \(N\) from 0 to \(c+d i\) (c) Find the equation of the line \(L\) through \(a+b i\) and parallel to line \(N\) of part (b). [Hint: The point \((a, b)\) is on \(L\) find the slope of \(L\) by using part (b) and facts about the slope of parallel lines.] (d) Find the equation of the line \(M\) through \(c+d i\) and parallel to line \(K\) of part (a). (e) Label the lines \(K, L, M,\) and \(N\) in the figure. (f) Show by using substitution that the point \((a+c, b+d)\) satisfies both the equation of line \(L\) and the equation of line \(M .\) Therefore, \((a+c, b+d)\) lies on both \(L\) and \(M\) since the only point on both \(L\) and \(M\) is the fourth vertex of the parallelogram (see the figure), this vertex must be \((a+c, b+d) .\) Hence, this vertex has coordinate $$(a+c)+(b+d) i=(a+b i)+(c+d i)$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=5, \theta=30^{\circ}$$

Find the component form of the vector \(v\) whose magnitude and direction angle \(\theta\) are given. $$\|\mathbf{v}\|=3, \boldsymbol{\theta}=310^{\circ}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\frac{\cos \pi+i \sin \pi}{\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}}$$

Find proju \(v\) and proju u. $$\mathbf{u}=2 \mathbf{i}-3 \mathbf{j}, \mathbf{v}=\mathbf{i}+2 \mathbf{j}$$
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