Question

find the distance between
-3 + 2i
and
5 + i
\sqrt{\boxed{?}}

Explanation:

Step1: Recall the distance formula for complex numbers

For two complex numbers \( z_1 = a + bi \) and \( z_2 = c + di \), the distance \( d \) between them is given by \( d=\sqrt{(a - c)^2+(b - d)^2} \). Here, \( z_1=-3 + 2i \) (so \( a=-3,b = 2 \)) and \( z_2 = 5 + i \) (so \( c = 5,d = 1 \)).

Step2: Calculate the differences in real and imaginary parts

First, find \( a - c=-3-5=-8 \) and \( b - d=2 - 1 = 1 \).

Step3: Square the differences and sum them

Square the differences: \( (-8)^2=64 \) and \( 1^2 = 1 \). Then sum them: \( 64+1=65 \).

Answer:

\( 65 \)