Find the distance between the complex numbers in the complex plane. 71, 4 - 3i
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Finding the Distance Between Complex Numbers on the Complex Plane**
To find the distance between the complex numbers on the complex plane, we are considering two given complex numbers: \(7i\) and \(4 - 3i\).
In this context, complex numbers can be treated as points in a two-dimensional plane, where the horizontal axis (x-axis) represents the real part of the number, and the vertical axis (y-axis) represents the imaginary part of the number.
Let's denote these points as:
- \(A(0, 7)\) for \(7i\)
- \(B(4, -3)\) for \(4 - 3i\)
The distance \(d\) between these two points can be found using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \]
Plugging in the coordinates of points \(A\) and \(B\):
- \(x_1 = 0\), \(y_1 = 7\)
- \(x_2 = 4\), \(y_2 = -3\)
\[ d = \sqrt{(4 - 0)^2 + (-3 - 7)^2 } \]
\[ d = \sqrt{4^2 + (-10)^2 } \]
\[ d = \sqrt{16 + 100} \]
\[ d = \sqrt{116} \]
\[ d = 2\sqrt{29} \]
Thus, the distance between the complex numbers \(7i\) and \(4 - 3i\) in the complex plane is \(2\sqrt{29}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9fe49cee-acc8-403b-8d12-20ec3514f583%2Fbac2bc9f-2827-4077-b01a-933ea5eb8ace%2Fzm898h_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Distance Between Complex Numbers on the Complex Plane**
To find the distance between the complex numbers on the complex plane, we are considering two given complex numbers: \(7i\) and \(4 - 3i\).
In this context, complex numbers can be treated as points in a two-dimensional plane, where the horizontal axis (x-axis) represents the real part of the number, and the vertical axis (y-axis) represents the imaginary part of the number.
Let's denote these points as:
- \(A(0, 7)\) for \(7i\)
- \(B(4, -3)\) for \(4 - 3i\)
The distance \(d\) between these two points can be found using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \]
Plugging in the coordinates of points \(A\) and \(B\):
- \(x_1 = 0\), \(y_1 = 7\)
- \(x_2 = 4\), \(y_2 = -3\)
\[ d = \sqrt{(4 - 0)^2 + (-3 - 7)^2 } \]
\[ d = \sqrt{4^2 + (-10)^2 } \]
\[ d = \sqrt{16 + 100} \]
\[ d = \sqrt{116} \]
\[ d = 2\sqrt{29} \]
Thus, the distance between the complex numbers \(7i\) and \(4 - 3i\) in the complex plane is \(2\sqrt{29}\).
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