Question

Find the distance using the modulus of the difference between z1 = -8 + 3i and z2 = 7 - 4i. Show all work for full credit.

Answer 1

Answer:

$\sqrt{274}$

Step-by-step explanation:

Given $z_1=-8+3i,\ z_2=7-4i.$

1. Find the difference $z_1-z_2:$

$z=z_1-z_2=-8+3i-(7-4i)=-8+3i-7+4i=(-8-7)+(3i+4i)=-15+7i.$

This complex number has real part $Rez=-15$ and imaginary part $Imz=7.$

2. The modulus of complex number z is

$|z|=\sqrt{Re^2z+Im^2z}=\sqrt{(-15)^2+7^2}=\sqrt{225+49}=\sqrt{274}.$

Answer 2

Answer:

The distance between $z_1\ \text{and}\ z_2$ is:

                        16.5529 units

Step-by-step explanation:

We know that the difference between two complex numbers:

$z_1=a_1+ib_1\ \text{and}\ z_2=a_2+ib_2$ is given by:

$|z_1-z_2|=|(a_1+ib_1)-(a_2+ib_2)|\\\\i.e.\\\\|z_1-z_2|=|(a_1-a_2)+i(b_1-b_2)|\\\\|z_1-z_2|=\sqrt{(a_1-a_2)^2+(b_1-b_2)^2}$

Here we have:

$z_1=-8+3i\ \text{and}\ z_2=7-4i$

i.e.

$a_1=-8\ ,\ b_1=3,\ a_2=7\ \text{and}\ b_2=-4$

i.e. we have:

$|z_1-z_2|=\sqrt{(-8-7)^2+(3-(-4))^2}\\\\|z_1-z_2|=\sqrt{15^2+7^2}\\\\|z_1-z_2|=\sqrt{225+49}\\\\|z_1-z_2|=\sqrt{274}\\\\|z_1-z_2|=16.5529$