Let $z_1 = 1 + 3i$ and $z_2 = 2 - i$, where $i^2 = -1$, are two complex numbers - Leaving Cert Mathematics - Question 2 - 2018

To plot the points:

• $z_1 = 1 + 3i$ is at (1, 3)
• $z_2 = 2 - i$ is at (2, -1)
• $z_3 = 3 + 2i$ is at (3, 2)

Label each point accordingly on the graph.

First, calculate $|z_2 - z_3|$:

$z_2 - z_3 = (2 - i) - (3 + 2i) = -1 - 3i$

Thus, $|z_2 - z_3| = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}$

Now calculate $|z_1 + z_2|$:

$z_1 + z_2 = (1 + 3i) + (2 - i) = 3 + 2i$

Thus, $|z_1 + z_2| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$

So, $|z_2 - z_3| \neq |z_1 + z_2|$.

To find $w$:

$w = \frac{z_1}{z_2} = \frac{1 + 3i}{2 - i}$

Multiply the numerator and denominator by the conjugate of the denominator:

$w = \frac{(1 + 3i)(2 + i)}{(2 - i)(2 + i)}$

Calculating the denominator:

$(2 - i)(2 + i) = 4 + 1 = 5$

Calculating the numerator:

$(1 + 3i)(2 + i) = 2 + i + 6i + 3i^2 = 2 + 7i - 3 = -1 + 7i$

Thus, $w = \frac{-1 + 7i}{5} = -\frac{1}{5} + \frac{7}{5}i$

So, $w = -\frac{1}{5} + \frac{7}{5}i$.