Question

Find the distance d between 21 = (-1+ 8i) and z2 = (-3 – 2i).Express your answer in exact terms and simplify, if needed.d=

Answer 1

The rule of the distance between two points is

$d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}$

Since the given points are

$z_1=(-1,8i),z_2=(-3,-2i)$

Then, let

$\begin{gathered} x_1=-1,x_2=-3 \\ y_1=8i,y_2=-2i \end{gathered}$

Substitute them in the rule above

$\begin{gathered} d=\sqrt[]{(-3-\lbrack-1\rbrack)^2+(-2i-8i)^2} \\ d=\sqrt[]{(-3+1)^2+(-10i)^2} \\ d=\sqrt[]{(-2)^2+(100i)^2} \end{gathered}$

Remember i^2 = -1, then

$\begin{gathered} d=\sqrt[]{4+(100i^2)} \\ i^2=-1 \\ d=\sqrt[]{4+100(-1)} \\ d=\sqrt[]{4-100} \\ d=\sqrt[]{-96} \end{gathered}$

Make the negative number represented by i

$\begin{gathered} d=\sqrt[]{96}\times\sqrt[]{-1} \\ \sqrt[]{96}=4\sqrt[]{6},\sqrt[]{-1}=i \\ d=4\sqrt[]{6}i \end{gathered}$

The distance between z1 and z2 is

$d=4\sqrt[]{6}i$