Problem 66 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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 \((6, -\pi)\).

Step 3: 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, \((1-i)(3-3i) = 6\).

Key Concepts

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’s 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.