To find the distance between the two complex numbers [tex]\(2 + 3i\)[/tex] and [tex]\(-8 + 14i\)[/tex], we use the concept of the distance between two points in the complex plane. This concept is analogous to finding the distance between two points in the Cartesian coordinate system.
Let's denote the two complex numbers as [tex]\(z_1 = 2 + 3i\)[/tex] and [tex]\(z_2 = -8 + 14i\)[/tex].
### Step-by-Step Solution:
1. Identify the Real and Imaginary Parts of Each Complex Number:
- For [tex]\(z_1 = 2 + 3i\)[/tex]:
- Real part: [tex]\(a_1 = 2\)[/tex]
- Imaginary part: [tex]\(b_1 = 3\)[/tex]
- For [tex]\(z_2 = -8 + 14i\)[/tex]:
- Real part: [tex]\(a_2 = -8\)[/tex]
- Imaginary part: [tex]\(b_2 = 14\)[/tex]
2. Calculate the Difference in the Real and Imaginary Parts:
- Difference in the real parts: [tex]\(\Delta a = a_2 - a_1 = -8 - 2 = -10\)[/tex]
- Difference in the imaginary parts: [tex]\(\Delta b = b_2 - b_1 = 14 - 3 = 11\)[/tex]
3. Formulate the Distance Formula:
The distance [tex]\(d\)[/tex] between two points [tex]\((a_1, b_1)\)[/tex] and [tex]\((a_2, b_2)\)[/tex] in the complex plane is given by:
[tex]\[
d = \sqrt{(\Delta a)^2 + (\Delta b)^2}
\][/tex]
Substituting the values we calculated:
[tex]\[
d = \sqrt{(-10)^2 + (11)^2}
\][/tex]
4. Perform the Squaring Operations:
[tex]\[
(-10)^2 = 100
\][/tex]
[tex]\[
(11)^2 = 121
\][/tex]
5. Add the Squared Differences:
[tex]\[
100 + 121 = 221
\][/tex]
6. Calculate the Square Root:
[tex]\[
\sqrt{221} = 14.866068747318506
\][/tex]
Therefore, the distance between the complex numbers [tex]\(2 + 3i\)[/tex] and [tex]\(-8 + 14i\)[/tex] is:
[tex]\[
14.866068747318506
\][/tex]
