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Chapter 10: Problem 16

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 1-\sqrt{3} i $$

Short Answer

Expert verified
Polar: 2(\cos(-\frac{\pi}{3}) + i\sin(-\frac{\pi}{3})), Exponential: 2 e^{-i \frac{\pi}{3}}.

Step by step solution

01

- Determine the real and imaginary parts

Let鈥檚 identify the real and imaginary parts of the complex number. Given the complex number is \(1 - \sqrt{3}i\). Here, the real part (a) is 1 and the imaginary part (b) is -\(\sqrt{3}\).
02

- Plot the number in the Complex Plane

Plot the point (1, -鈭�3) on the complex plane. This corresponds to moving 1 unit to the right on the real axis and \(-\sqrt{3}\) units downward on the imaginary axis.
03

- Calculate the modulus (r)

The modulus of the complex number is calculated as: \[ r = \sqrt{a^2 + b^2} \] Substituting the values, we get: \[ r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]
04

- Calculate the argument (胃)

The argument is calculated using the arctangent function: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] Substituting the values, we get: \[ \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3}) \]. Since the number is in the fourth quadrant, 胃 = \(-\frac{\pi}{3}\).
05

- Write the polar form

The polar form of a complex number is given by: \[ z = r (\cos \theta + i \sin \theta) \] Substituting the values, we get: \[ z = 2 (\cos(-\frac{\pi}{3}) + i \sin(-\frac{\pi}{3})) \]
06

- Write the exponential form

The exponential form of a complex number is given by: \[ z = r e^{i\theta} \] Substituting the values, we get: \[ z = 2 e^{-i\frac{\pi}{3}} \]

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Key Concepts

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

Complex Plane
The complex plane is a two-dimensional plane. Here, a complex number is represented graphically. The horizontal axis represents the real part of the number while the vertical axis represents the imaginary part. For example, the number \(1 - \sqrt{3}i\) can be plotted by moving 1 unit to the right (along the real axis) and \(-\sqrt{3}\) units downwards (along the imaginary axis).

  • Real Axis: Horizontal axis
  • Imaginary Axis: Vertical axis
  • Coordinate: (Real, Imaginary)
Polar Form
Polar form expresses a complex number in terms of modulus and argument. This form is very useful in understanding the magnitude and direction of a complex number.

  • Modulus (r): Distance from origin
  • Argument (胃): Angle with the positive real axis
For the number \(1 - \sqrt{3}i\), we first calculate the modulus as \(r = 2\) and then the argument as \(\theta = -\frac{\pi}{3}\). The polar form is then: \[ z = 2(\cos(-\frac{\pi}{3}) + i \sin(-\frac{\pi}{3})) \]
Exponential Form
The exponential form of a complex number is another way to represent it, using Euler's formula. It ties together the polar form with an exponent involving the imaginary unit, i.

  • Formula: \(z = re^{i\theta}\)
  • Relationship: Links the magnitude and phase angle
For our example, with modulus 2 and argument \(-\frac{\pi}{3}\), the exponential form becomes: \[ z = 2e^{-i\frac{\pi}{3}} \]
Modulus
The modulus of a complex number is its distance from the origin on the complex plane. It's calculated using the Pythagorean theorem: \[ r = \sqrt{a^2 + b^2} \].

  • For \(a = 1\) and \(b = -\sqrt{3}\): \( r = 2 \)
  • Interpretation: Magnitude of the number
Argument
The argument of a complex number is the angle it makes with the positive real axis. It's calculated using the arctangent function: \[ \theta = \tan^{-1}(\frac{b}{a}) \].

  • Placement: Fourth quadrant
  • Value: \(\theta = -\frac{\pi}{3}\)
It helps in determining the direction of the complex number in the plane.

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Most popular questions from this chapter

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=-\mathbf{i}-5 \mathbf{j}\)

Let \(\mathbf{v}\) and \(\mathbf{w}\) denote two nonzero vectors. Show that the vector \(\mathbf{v}-\alpha \mathbf{w}\) is orthogonal to \(\mathbf{w}\) if \(\alpha=\frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^{2}}\)

Find \(a\) so that the vectors \(\mathbf{v}=\mathbf{i}-a \mathbf{j}\) and \(\mathbf{w}=2 \mathbf{i}+3 \mathbf{j}\) are orthogonal.

Radar station \(A\) uses a coordinate system where \(A\) is located at the pole and due east is the polar axis. On this system, two other radar stations, \(B\) and \(C,\) are located at coordinates \(\left(150,-24^{\circ}\right)\) and \(\left(100,32^{\circ}\right)\) respectively. If radar station \(B\) uses a coordinate system where \(B\) is located at the pole and due east is the polar axis, then what are the coordinates of radar stations \(A\) and \(C\) on this second system? Round answers to one decimal place.

Suppose that \(\mathbf{v}\) and \(\mathbf{w}\) are unit vectors. If the angle between \(\mathbf{v}\) and \(\mathbf{i}\) is \(\alpha\) and the angle between \(\mathbf{w}\) and \(\mathbf{i}\) is \(\beta\), use the idea of the dot product \(\mathbf{v} \cdot \mathbf{w}\) to prove that $$ \cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta $$
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