Problem 34 Find the modulus and argument (a... [FREE SOLUTION]

Chapter 10: Problem 34

Find the modulus and argument (amplitude) of each of the complex numbers $$z_{1}=2 \sqrt{3}-2 i, z_{2}=2+2 i \text { and } z_{3}=(2 \sqrt{3}-2 i)(2+2 i)$$

Short Answer

Expert verified

The moduli of $ z_1, z_2, z_3 $ are 4, $ 2\sqrt{2} $, and $ 4\sqrt{2} $. The arguments are $ -\frac{\pi}{6} $, $ \frac{\pi}{4} $, and $ \frac{\pi}{2} $ respectively.

Step by step solution

Identify the Complex Number Components

The complex numbers given are $ z_1 = 2\sqrt{3} - 2i $ and $ z_2 = 2 + 2i $. For each complex number $ z = a + bi $, where $ a $ is the real part and $ b $ is the imaginary part.

Calculate Modulus of z1

The modulus of a complex number $ z = a + bi $ is given by $ |z| = \sqrt{a^2 + b^2} $. For $ z_1 = 2\sqrt{3} - 2i $, the modulus is $ |z_1| = \sqrt{ (2\sqrt{3})^2 + (-2)^2 } = \sqrt{12 + 4} = \sqrt{16} = 4 $.

Calculate Argument of z1

The argument $ \theta $ of a complex number $ z = a + bi $ is $ \theta = \tan^{-1}(\frac{b}{a}) $. For $ z_1 = 2\sqrt{3} - 2i $, $ \theta = \tan^{-1}\left(\frac{-2}{2\sqrt{3}}\right) = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} $.

Calculate Modulus of z2

Using the same formula for modulus, for $ z_2 = 2 + 2i $, the modulus is $ |z_2| = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} $.

Calculate Argument of z2

For $ z_2 = 2 + 2i $, the argument $ \theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4} $.

Calculate Product z3

For $ z_3 = (2\sqrt{3} - 2i)(2 + 2i) $, distribute the terms: \[(2\sqrt{3} \cdot 2 + 2\sqrt{3} \cdot 2i - 2 \cdot 2 - 2 \cdot 2i) = (4\sqrt{3} + 4\sqrt{3}i - 4 - 4i) = 4\sqrt{3} - 4 + (4\sqrt{3} - 4)i)\]Calculating further: The real part is $ 4\sqrt{3} - 4 $The imaginary part is $ 4\sqrt{3} - 4 $.

Calculate Modulus of z3

Using the modulus formula on $ z_3 $, we have:\[|z_3| = \sqrt{(4\sqrt{3} - 4)^2 + (4\sqrt{3} - 4)^2}\]Simplifying, this expresses to:\[= \sqrt{2(4\sqrt{3} - 4)^2} = \sqrt{2(48 + 32 - 32\sqrt{3})}\]\[= \sqrt{160 - 64\sqrt{3}} = 4\sqrt{10 - 4\sqrt{3}}\]

Calculate Argument of z3

For argument of $ z_3 $, since both real and imaginary parts are equal, the argument is 45掳 or equivalently $ \frac{\pi}{4} $ radians, plus $ 180 $ degrees if necessary for correct quadrants.

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Key Concepts

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

Modulus of a Complex Number

The modulus of a complex number is a measure of its "size" or "length" in the complex plane. A complex number is generally written in the form $ z = a + bi $, where $ a $ is the real part and $ b $ is the imaginary part. To find the modulus, we use the formula:

$ |z| = \sqrt{a^2 + b^2} $

This formula is derived from the Pythagorean theorem since the complex plane can be seen as a coordinate system. For example, the modulus of $ z_1 = 2\sqrt{3} - 2i $ is calculated as:

$ |z_1| = \sqrt{(2\sqrt{3})^2 + (-2)^2} = \sqrt{12 + 4} = \sqrt{16} = 4 $

The modulus tells us how far the point representing the complex number is from the origin $(0,0)$ on a Cartesian coordinate plane. Higher values indicate further distance.

Argument of a Complex Number

The argument of a complex number is the angle formed with the positive direction of the x-axis (real axis), and is crucial for fully understanding the placement of the complex number in the complex plane. For a complex number written as $ z = a + bi $, the argument $ \theta $ is given by:

$ \theta = \tan^{-1}\left(\frac{b}{a}\right) $

The argument can be visualized as the angle you would turn from the positive x-axis to reach the line representing the complex number. For instance, with $ z_2 = 2 + 2i $, we calculate:

$ \theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = \frac{\pi}{4} $

The argument has to be adjusted based on the quadrant in which the point lies to ensure it's measured correctly. It's usually expressed in radians, with adjustments made for full rotations when necessary.

Step-by-Step Mathematical Solution

Solving for the modulus and argument of a complex number can be done systematically. Each step builds on the previous one:

Identify Components: Clearly label the real part $ a $ and imaginary part $ b $ from the form $ z = a + bi $.
Calculate Modulus: Use the formula $ |z| = \sqrt{a^2 + b^2} $ to find the modulus. Check each calculation for accuracy.
Determine Argument: Apply the formula $ \theta = \tan^{-1}\left(\frac{b}{a}\right) $ to find the angle. Adjust for the correct quadrant.
Handle Complex Operations: When dealing with products like $ z_3 = (2\sqrt{3} - 2i)(2 + 2i) $, distribute and simplify before finding modulus and argument.

Always pause to verify calculations to ensure no mistakes from arithmetic errors or misinterpretations, particularly in trigonometry and when the result is used in further calculations.

91影视

Find the modulus and argument (amplitude) of each of the complex numbers $$z_{1}=2 \sqrt{3}-2 i, z_{2}=2+2 i \text { and } z_{3}=(2 \sqrt{3}-2 i)(2+2 i)$$

Short Answer

Step by step solution

Identify the Complex Number Components

Calculate Modulus of z1

Calculate Argument of z1

Calculate Modulus of z2

Calculate Argument of z2

Calculate Product z3

Calculate Modulus of z3

Calculate Argument of z3

Key Concepts

Modulus of a Complex Number

Argument of a Complex Number

Step-by-Step Mathematical Solution

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