Question

given $z_1$ and $z_2$, find the distance between them.
$z_1 = 3 + 7i$ and $z_2 = -5 - 2i$
$|z_1 - z_2| = \sqrt{?}$

Explanation:

Step1: Calculate \( z_1 - z_2 \)

Substitute \( z_1 = 3 + 7i \) and \( z_2 = -5 - 2i \) into \( z_1 - z_2 \):
\( z_1 - z_2=(3 + 7i)-(-5 - 2i)=3 + 7i + 5 + 2i=(3 + 5)+(7i+2i)=8 + 9i \)

Step2: Calculate the modulus of \( z_1 - z_2 \)

The modulus of a complex number \( a+bi \) is \( \sqrt{a^{2}+b^{2}} \). For \( 8 + 9i \), \( a = 8 \), \( b = 9 \), so \( |z_1 - z_2|=\sqrt{8^{2}+9^{2}}=\sqrt{64 + 81}=\sqrt{145} \). We need to find the value inside the square root, which is \( 8^{2}+9^{2}=64 + 81 = 145 \)

Answer:

145