Problem 8
Question
If \(r=2\) and \(0 \leq \theta \leq \frac{\pi}{2}, w=\sqrt{2} e^{i \theta / 2} .\) Therefore \(|w|=\sqrt{2}\) and \(0 \leq \operatorname{Arg} w \leq \pi / 4 .\) This image is a circular arc.
Step-by-Step Solution
Verified Answer
The image is a circular arc of radius \( \sqrt{2} \), subtending an angle from \( 0 \) to \( \frac{\pi}{4} \).
1Step 1: Understand the Given Parameters
The problem states that \( r = 2 \) and the angle \( \theta \) ranges from \( 0 \) to \( \frac{\pi}{2} \). It defines \( w = \sqrt{2} e^{i \theta / 2} \). This means \( w \) is a complex number in polar form.
2Step 2: Identify the Magnitude of w
From the given formula \( w = \sqrt{2} e^{i \theta / 2} \), we see that the magnitude of \( w \), denoted as \(|w|\), is directly the coefficient \( \sqrt{2} \). Hence, \(|w| = \sqrt{2} \).
3Step 3: Find the Argument of w
The argument of \( w \), \( \operatorname{Arg}(w) \), is given by \( \frac{\theta}{2} \) because \( w = \sqrt{2} e^{i \theta / 2} \). Since \( \theta \) ranges from \( 0 \) to \( \frac{\pi}{2} \), \( \operatorname{Arg}(w) \) will range from \( 0 \) to \( \frac{\pi}{4} \).
4Step 4: Determine the Path of w
\( w \) represents a point on a complex plane that traces an arc. With \( |w| = \sqrt{2} \) fixed, the point travels along a circular arc, because changing \( \theta \) changes the angle \( \operatorname{Arg}(w) \), tracing out an arc of a circle centered at the origin with radius \( \sqrt{2} \).
Key Concepts
Polar FormMagnitude of a Complex NumberArgument of a Complex Number
Polar Form
The polar form of a complex number provides a different yet valuable representation compared to its standard rectangular form. A complex number can be expressed as either \( a + bi \), where \( a \) and \( b \) are real numbers, or in polar form \( r e^{i\theta} \). Here, \( r \) is the magnitude of the complex number, and \( \theta \) is the argument, or angle.
Polar form is particularly useful in these scenarios:
Polar form is particularly useful in these scenarios:
- Multiplying or dividing complex numbers becomes straightforward, as you multiply/divide magnitudes and add/subtract angles.
- It provides a visual insight when dealing with complex numbers, especially when graphing on the complex plane.
Magnitude of a Complex Number
The magnitude, also known as the modulus, of a complex number is a measure of its size or length from the origin in the complex plane. For any complex number \( z = a + bi \), the magnitude \( |z| \) is calculated using the formula \( \sqrt{a^2 + b^2} \).
In polar form, the magnitude is denoted directly by \( r \). This means that for a complex number in the form \( z = r e^{i\theta} \), the magnitude is \( r \). In our specific example, since \( w = \sqrt{2} e^{i \theta/2} \), the magnitude \( |w| \) is simply \( \sqrt{2} \). This shows the distance of \( w \) from the origin in the complex plane, unaffected by the angle \( \theta/2 \).
Understanding the magnitude is crucial when analyzing how complex numbers behave geometrically, especially when considering transformations such as rotations and scaling.
In polar form, the magnitude is denoted directly by \( r \). This means that for a complex number in the form \( z = r e^{i\theta} \), the magnitude is \( r \). In our specific example, since \( w = \sqrt{2} e^{i \theta/2} \), the magnitude \( |w| \) is simply \( \sqrt{2} \). This shows the distance of \( w \) from the origin in the complex plane, unaffected by the angle \( \theta/2 \).
Understanding the magnitude is crucial when analyzing how complex numbers behave geometrically, especially when considering transformations such as rotations and scaling.
Argument of a Complex Number
The argument of a complex number is the angle it makes with the positive real axis on the complex plane. It essentially shows the direction of the complex number. For any complex number represented in polar form \( z = r e^{i\theta} \), the argument \( \theta \) can be found directly as it is the angle part.
To determine the argument of a complex number \( z = a + bi \), the formula \( \operatorname{Arg}(z) = \tan^{-1}(\frac{b}{a}) \) is often used. However, in the scenario where the complex number is already given in polar form, finding the argument becomes a matter of identifying the angle component in the expression.
In our example, the complex number is \( w = \sqrt{2} e^{i \theta / 2} \), the argument is \( \operatorname{Arg}(w) = \frac{\theta}{2} \). Given that \( \theta \) ranges from \( 0 \) to \( \frac{\pi}{2} \), this implies that \( \operatorname{Arg}(w) \) will range between \( 0 \) and \( \frac{\pi}{4} \).
Understanding the argument helps in visualizing and discussing rotations of complex numbers in the plane.
To determine the argument of a complex number \( z = a + bi \), the formula \( \operatorname{Arg}(z) = \tan^{-1}(\frac{b}{a}) \) is often used. However, in the scenario where the complex number is already given in polar form, finding the argument becomes a matter of identifying the angle component in the expression.
In our example, the complex number is \( w = \sqrt{2} e^{i \theta / 2} \), the argument is \( \operatorname{Arg}(w) = \frac{\theta}{2} \). Given that \( \theta \) ranges from \( 0 \) to \( \frac{\pi}{2} \), this implies that \( \operatorname{Arg}(w) \) will range between \( 0 \) and \( \frac{\pi}{4} \).
Understanding the argument helps in visualizing and discussing rotations of complex numbers in the plane.
Other exercises in this chapter
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View solution Problem 7
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View solution Problem 9
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View solution Problem 10
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View solution