Warner Gibson
04/01/2024 · Middle School
QUESTION 1 1.1 Given: \( Z_{1}=5-4 i, Z_{2}=3-i \) and \( Z_{3}=-3 i \) Calculate the following and leave your answers in the form \( a+b i \) : 1.1.1 \( Z_{1}+Z_{2} \) \( 1.1 .2 Z_{1} \times Z_{2} \) \( 1.1 .3 \frac{Z_{3}}{Z_{1}} \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To solve the given problems, we will perform the necessary operations on the complex numbers \( Z_1 = 5 - 4i \), \( Z_2 = 3 - i \), and \( Z_3 = -3i \).
### 1.1.1 \( Z_{1} + Z_{2} \)
To add the complex numbers, we add their real parts and their imaginary parts separately:
\[
Z_{1} + Z_{2} = (5 - 4i) + (3 - i) = (5 + 3) + (-4 - 1)i = 8 - 5i
\]
So,
\[
Z_{1} + Z_{2} = 8 - 5i
\]
### 1.1.2 \( Z_{1} \times Z_{2} \)
To multiply the complex numbers, we use the distributive property (FOIL method):
\[
Z_{1} \times Z_{2} = (5 - 4i)(3 - i)
\]
Calculating this step-by-step:
\[
= 5 \cdot 3 + 5 \cdot (-i) - 4i \cdot 3 - 4i \cdot (-i)
\]
\[
= 15 - 5i - 12i + 4i^2
\]
Since \( i^2 = -1 \), we have:
\[
= 15 - 5i - 12i + 4(-1)
\]
\[
= 15 - 4 - 17i
\]
\[
= 11 - 17i
\]
So,
\[
Z_{1} \times Z_{2} = 11 - 17i
\]
### 1.1.3 \( \frac{Z_{3}}{Z_{1}} \)
To divide the complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator:
\[
\frac{Z_{3}}{Z_{1}} = \frac{-3i}{5 - 4i} \cdot \frac{5 + 4i}{5 + 4i}
\]
Calculating the denominator:
\[
(5 - 4i)(5 + 4i) = 5^2 - (4i)^2 = 25 - 16(-1) = 25 + 16 = 41
\]
Now calculating the numerator:
\[
-3i(5 + 4i) = -15i - 12i^2 = -15i + 12 = 12 - 15i
\]
Putting it all together:
\[
\frac{Z_{3}}{Z_{1}} = \frac{12 - 15i}{41} = \frac{12}{41} - \frac{15}{41}i
\]
So,
\[
\frac{Z_{3}}{Z_{1}} = \frac{12}{41} - \frac{15}{41}i
\]
### Summary of Answers
1.1.1 \( Z_{1} + Z_{2} = 8 - 5i \)
1.1.2 \( Z_{1} \times Z_{2} = 11 - 17i \)
1.1.3 \( \frac{Z_{3}}{Z_{1}} = \frac{12}{41} - \frac{15}{41}i \)
Quick Answer
1.1.1 \( Z_{1} + Z_{2} = 8 - 5i \)
1.1.2 \( Z_{1} \times Z_{2} = 11 - 17i \)
1.1.3 \( \frac{Z_{3}}{Z_{1}} = \frac{12}{41} - \frac{15}{41}i \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit