Question

given the following two complex numbers find the distance between them.
z1 = - 12 - 17i
z2 = -6 - 12i

Answer 1

Answer:

$\sqrt{61}$

Step-by-step explanation:

The distance between two complex numbers in the form z = a + bi can be found using the distance formula:

$\boxed{\begin{array}{l}\underline{\sf Distance \;Formula\; (complex \;numbers)}\\\\d=\sqrt{(a_2-a_1)^2+(b_2-b_1)^2}\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\;d\;\textsf{is the distance between two complex numbers.} \\\phantom{ww}\bullet\;\;\textsf{ (a_1+b_1i) and (a_2+b_2i) are the two complex numbers.}\end{array}}$

Given that the two complex numbers are z₁ = -12 - 17i and z₂ = -6 - 12i, then:

$a_1=-12$

$b_1=-17$

$a_2=-6$

$b_2=-12$

Substitute these values into the distance formula:

$\begin{aligned}d&=\sqrt{(-6-(-12))^2+(-12-(-17))^2}\\\\d&=\sqrt{(-6+12)^2+(-12+17)^2}\\\\d&=\sqrt{(6)^2+(5)^2}\\\\d&=\sqrt{36+25}\\\\d&=\sqrt{61}\end{aligned}$

Therefore, the exact distance between the complex numbers z₁ = -12 - 17i and z₂ = -6 - 12i is:

$\huge\boxed{\boxed{\sqrt{61}}}$

The distance is 7.81 rounded to two decimal places.