The complex number $z_1 = a + bi$, where $i^2 = -1$, is shown on the Argand diagram below - Leaving Cert Mathematics - Question 6 - 2015

After reflection in the real axis, the $y$-coordinate changes sign while the $x$-coordinate remains the same. Thus:

c = 3

d = -1

We can find $\cos \theta$ using the formula:

$|z_1| = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10}$

From the triangle formed, we have:

$\cos \theta = \frac{10}{\sqrt{10} \cdot \sqrt{10}} = \frac{10}{10} - 4$

Calculating gives:

$\cos \theta = 0.8$

From previous parts:

• $|z_1| = \sqrt{10}$
• $c = 3$, $d = -1$, and $|z_3| = |0 + 0i|$

Thus:

$|z_1| |z_2| \cos \theta = \sqrt{10} \cdot |3| \cdot 0.8 = 8$

Now, evaluating $ac + bd$:

$ac + bd = 3 \cdot 3 + 1 \cdot (-1) = 9 - 1 = 8$

Hence, the equality holds true.