To find the distance between the complex numbers [tex]\(-2 - 3i\)[/tex] and [tex]\(3 + 9i\)[/tex], we can use the distance formula for complex numbers. The distance [tex]\(d\)[/tex] between two complex numbers [tex]\(z_1 = a + bi\)[/tex] and [tex]\(z_2 = c + di\)[/tex] is given by the formula:
[tex]\[ d = \sqrt{(c - a)^2 + (d - b)^2} \][/tex]
Let's denote the given complex numbers as:
[tex]\[ z_1 = -2 - 3i \][/tex]
[tex]\[ z_2 = 3 + 9i \][/tex]
Here, [tex]\(a = -2\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = 3\)[/tex], and [tex]\(d = 9\)[/tex]. Substitute these values into the distance formula:
[tex]\[ d = \sqrt{(3 - (-2))^2 + (9 - (-3))^2} \][/tex]
Simplify the terms inside the square root:
[tex]\[ c - a = 3 - (-2) = 3 + 2 = 5 \][/tex]
[tex]\[ d - b = 9 - (-3) = 9 + 3 = 12 \][/tex]
Now square these differences:
[tex]\[ (c - a)^2 = 5^2 = 25 \][/tex]
[tex]\[ (d - b)^2 = 12^2 = 144 \][/tex]
Then add these squared values:
[tex]\[ d = \sqrt{25 + 144} \][/tex]
[tex]\[ d = \sqrt{169} \][/tex]
Finally, take the square root of the sum:
[tex]\[ d = 13 \][/tex]
Therefore, the distance between the complex numbers [tex]\(-2 - 3i\)[/tex] and [tex]\(3 + 9i\)[/tex] is [tex]\(13.0\)[/tex].
