Question

given z₁ and z₂, find the distance between them.
z₁ = 3 + 16i and z₂ = -6 + 4i
|z₁ - z₂| = ?

Explanation:

Step1: Subtract \( z_2 \) from \( z_1 \)

First, we calculate \( z_1 - z_2 \). Given \( z_1 = 3 + 16i \) and \( z_2 = -6 + 4i \), we have:
\[

$$\begin{align*} z_1 - z_2&=(3 + 16i)-(-6 + 4i)\\ &=3 + 16i + 6 - 4i\\ &=(3 + 6)+(16i - 4i)\\ &=9 + 12i \end{align*}$$

\]

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

The modulus of a complex number \( a + bi \) is given by \( \sqrt{a^2 + b^2} \). For \( z_1 - z_2 = 9 + 12i \), where \( a = 9 \) and \( b = 12 \), we calculate the modulus:
\[

$$\begin{align*} |z_1 - z_2|&=\sqrt{9^2 + 12^2}\\ &=\sqrt{81 + 144}\\ &=\sqrt{225}\\ & = 15 \end{align*}$$

\]

Answer:

\( 15 \)