Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Finding the Distance Between Complex Numbers on the Complex Plane
**Problem Statement:**
Find the distance between the complex numbers in the complex plane.
\[ 1 + 4i , \quad -1 + 7i \]
**Solution:**
The distance \(d\) between two complex numbers \(z_1 = x_1 + y_1i\) and \(z_2 = x_2 + y_2i\) in the complex plane is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For the complex numbers \(z_1 = 1 + 4i\) and \(z_2 = -1 + 7i\):
- \(x_1 = 1\)
- \(y_1 = 4\)
- \(x_2 = -1\)
- \(y_2 = 7\)
**Steps:**
1. Compute the difference between the real parts: \((x_2 - x_1) = (-1 - 1) = -2\)
2. Compute the difference between the imaginary parts: \((y_2 - y_1) = (7 - 4) = 3\)
3. Substitute into the distance formula:
\[ d = \sqrt{(-2)^2 + (3)^2} \]
\[ d = \sqrt{4 + 9} \]
\[ d = \sqrt{13} \]
Thus, the distance between the complex numbers \(1 + 4i\) and \(-1 + 7i\) is \(\sqrt{13}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fab864d-7419-4b8c-bc50-25971d273c3c%2F42df9acf-cc6c-44a9-97d4-b33b3ced13bb%2F9yydtc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Distance Between Complex Numbers on the Complex Plane
**Problem Statement:**
Find the distance between the complex numbers in the complex plane.
\[ 1 + 4i , \quad -1 + 7i \]
**Solution:**
The distance \(d\) between two complex numbers \(z_1 = x_1 + y_1i\) and \(z_2 = x_2 + y_2i\) in the complex plane is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For the complex numbers \(z_1 = 1 + 4i\) and \(z_2 = -1 + 7i\):
- \(x_1 = 1\)
- \(y_1 = 4\)
- \(x_2 = -1\)
- \(y_2 = 7\)
**Steps:**
1. Compute the difference between the real parts: \((x_2 - x_1) = (-1 - 1) = -2\)
2. Compute the difference between the imaginary parts: \((y_2 - y_1) = (7 - 4) = 3\)
3. Substitute into the distance formula:
\[ d = \sqrt{(-2)^2 + (3)^2} \]
\[ d = \sqrt{4 + 9} \]
\[ d = \sqrt{13} \]
Thus, the distance between the complex numbers \(1 + 4i\) and \(-1 + 7i\) is \(\sqrt{13}\).
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