Question

given $z_1$ and $z_2$, find the distance between them.
$z_1 = 6 + 3i$ and $z_2 = 2 - 5i$
$|z_1 - z_2| = ?\sqrt{}$

Explanation:

Step1: Calculate \( z_1 - z_2 \)

Subtract the complex numbers \( z_1 = 6 + 3i \) and \( z_2 = 2 - 5i \).
\( z_1 - z_2=(6 + 3i)-(2 - 5i)=6 + 3i - 2 + 5i=(6 - 2)+(3i + 5i)=4 + 8i \)

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

The modulus of a complex number \( a+bi \) is \( \sqrt{a^{2}+b^{2}} \). For \( 4 + 8i \), \( a = 4 \), \( b = 8 \).
\( |z_1 - z_2|=\sqrt{4^{2}+8^{2}}=\sqrt{16 + 64}=\sqrt{80}=\sqrt{16\times5}=4\sqrt{5} \)

Answer:

\( 4\sqrt{5} \) (So the first box is \( 4 \) and the second box is \( 5 \))