Problem 17 - Algebra and Trigonometry

Question

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ -3 i $$

Step-by-Step Solution

Answer

The complex number \( -3i \) in polar form is \( 3 (\text{cos}(-\frac{π}{2}) + i \text{sin}(-\frac{π}{2})) \) and in exponential form is \( 3 e^{-i\frac{π}{2}} \).

Step 1 - Identify the Complex Number

Given complex number: \( -3i \). This number has no real part and its imaginary part is \( -3 \).

Step 2 - Plot the Complex Number

To plot \( -3i \) on the complex plane, note that it lies on the imaginary axis. The point is at (0, -3).

Step 3 - Convert to Polar Form

The polar form of a complex number is given by \( r(\text{cos} \theta + i\text{sin} \theta) \) where \( r \) is the modulus and \( \theta \) is the argument.1. Modulus: \( r = | -3i | = 3 \).2. Argument: \( \theta = \text{arg}(-3i) = -\frac{\text{π}}{2} \).So, the polar form is \( 3 (\text{cos}(-\frac{π}{2}) + i \text{sin}(-\frac{π}{2})) \).

Step 4 - Convert to Exponential Form

The exponential form of a complex number is given by \( r e^{i\theta} \) where \( r \) is the modulus and \( \theta \) is the argument.Given: \( r = 3 \) and \( \theta = -\frac{π}{2} \).So, the exponential form is \( 3 e^{-i\frac{π}{2}} \).

Key Concepts

complex plane

The complex plane is a two-dimensional plane used to represent complex numbers visually. Each complex number has a real part and an imaginary part. We plot the real part on the horizontal axis (real axis) and the imaginary part on the vertical axis (imaginary axis). For example, the number \( -3i \) has no real part and an imaginary part of \( -3 \). Therefore, it is plotted at the point (0, -3) on the complex plane. Visualizing complex numbers helps you understand their behavior better.

polar form

The polar form of a complex number expresses the number in terms of its magnitude (modulus) and angle (argument). Given a complex number \( z = x + yi \), you can convert it to polar form \( r(\text{cos} \theta + i\text{sin} \theta) \). Here, \( r \) is the distance from the origin to the point \( (x, y) \) and \( \theta \) is the angle with the positive real axis. For \( -3i \), the modulus \( r = 3 \) and argument \( \theta = -\frac{\text{π}}{2} \). Therefore, its polar form is \( 3(\text{cos}(-\frac{π}{2}) + i\text{sin}(-\frac{π}{2})) \).

exponential form

The exponential form of a complex number is a compact way to represent it using Euler's formula. Euler's formula states that \( e^{i \theta} = \text{cos} \theta + i\text{sin} \theta \). Hence, a complex number in polar form \( r(\text{cos} \theta + i \text{sin} \theta) \) can be written as \( r e^{i \theta} \). For our specific complex number \( -3i \), where \( r = 3 \) and \( \theta = -\frac{\text{π}}{2} \), the exponential form is \( 3 e^{-i \frac{π}{2}} \). This form is particularly useful in advanced mathematical analysis.

modulus

The modulus of a complex number \( z = x + yi \) represents its magnitude or distance from the origin in the complex plane. It is calculated using the formula \( |z| = \text{sqrt}(x^2 + y^2) \). For the complex number \( -3i \), there is no real part (\( x = 0 \)) and the imaginary part is \( -3 \). Thus, the modulus is \( | -3i | = \text{sqrt}(0^2 + (-3)^2) = 3 \). The modulus gives you a sense of how large the complete number is.

argument of complex number

The argument of a complex number is the angle formed with the positive real axis, measured in radians. It is denoted by \( \theta \). To find the argument of \( z = x + yi \), you can use trigonometry: \( \theta = \text{atan2}(y, x) \). For \( -3i \), the point lies on the negative imaginary axis, so \( \theta = -\frac{\text{π}}{2} \). The argument helps to determine the direction of the point from the origin on the complex plane.