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

`√(274)`

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|=√(Re^2z+Im^2z)=√((-15)^2+7^2)=√(225+49)=√(274).`

Answer 2

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

16.5529 units

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|=√((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|=√((-8-7)^2+(3-(-4))^2)\\\\|z_1-z_2|=√(15^2+7^2)\\\\|z_1-z_2|=√(225+49)\\\\|z_1-z_2|=√(274)\\\\|z_1-z_2|=16.5529`