Question

given $z_1$ and $z_2$, find the distance between them. $z_1 = 6 - 3i$ and $z_2 = -4 + 2i$ $|z_1 - z_2| = ?\sqrt{}$

Explanation:

Step1: Calculate \( z_1 - z_2 \)

Substitute \( z_1 = 6 - 3i \) and \( z_2 = -4 + 2i \) into \( z_1 - z_2 \):
\( z_1 - z_2=(6 - 3i)-(-4 + 2i)=6 - 3i + 4 - 2i=(6 + 4)+(-3i-2i)=10 - 5i \)

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

The modulus of a complex number \( a+bi \) is \( \sqrt{a^2 + b^2} \). For \( 10 - 5i \), \( a = 10 \), \( b=- 5 \):
\( |z_1 - z_2|=\sqrt{10^2+(-5)^2}=\sqrt{100 + 25}=\sqrt{125}=\sqrt{25\times5}=5\sqrt{5} \)

Answer:

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