Ejercicios resueltos de módulo y conjugado

$|z_1| = |5 - 2i| = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}$

$|z_2| = |-3 + 4i| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

$\overline{z_1} = \overline{5 - 2i} = 5 + 2i$

$\overline{z_2} = \overline{-3 + 4i} = -3 - 4i$

$z_1 \times \overline{z_1} = (5 - 2i)(5 + 2i) = 25 - 4i^2 = 25 + 4 = 29 = |z_1|^2$

$z_2 \times \overline{z_2} = (-3 + 4i)(-3 - 4i) = 9 - 16i^2 = 9 + 16 = 25 = |z_2|^2$

$\overline{z_1} + \overline{z_2} = (5 + 2i) + (-3 - 4i) = 2 - 2i$

$|\overline{z_1} + \overline{z_2}| = |2 - 2i| = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$

$z_1 + z_2 = (1 + 5i) + (6 - 3i) = 7 + 2i$

$|z_1 + z_2| = |7 + 2i| = \sqrt{49 + 4} = \sqrt{53} \approx 7.28$

$|z_1| + |z_2| = \sqrt{1 + 25} + \sqrt{36 + 9} = \sqrt{26} + \sqrt{45} = \sqrt{26} + 3\sqrt{5} \approx 5.10 + 6.71 = 11.81$

Se cumple: $|z_1 + z_2| \leq |z_1| + |z_2|$ ✓

$z_1z_2 = (1 + 5i)(6 - 3i) = 6 - 3i + 30i - 15i^2 = 6 + 27i + 15 = 21 + 27i$

$|z_1z_2| = |21 + 27i| = \sqrt{441 + 729} = \sqrt{1170} = 3\sqrt{130}$

$|z_1||z_2| = \sqrt{26} \cdot 3\sqrt{5} = 3\sqrt{130}$ ✓

$\overline{z_3} = \overline{-2 - 7i} = -2 + 7i$

$|\overline{z_3}| = |-2 + 7i| = \sqrt{4 + 49} = \sqrt{53}$

$|z_3| = |-2 - 7i| = \sqrt{4 + 49} = \sqrt{53}$

Se cumple: $|\overline{z_3}| = |z_3|$ ✓

$z_1z_2z_3 = (21 + 27i)(-2 - 7i) = -42 - 147i - 54i - 189i^2 = -42 - 201i + 189 = 147 - 201i$

$\overline{z_1z_2z_3} = \overline{147 - 201i} = 147 + 201i$

$\overline{z_1}\overline{z_2}\overline{z_3} = (1 - 5i)(6 + 3i)(-2 + 7i)$

$= (6 + 3i - 30i - 15i^2)(-2 + 7i) = (21 - 27i)(-2 + 7i) = -42 + 147i + 54i - 189i^2 = 147 + 201i$ ✓

Para que $z\overline{u} = u$, necesitamos $z = \frac{u}{\overline{u}}$

$\overline{u} = 3 - 4i$, por tanto $z = \frac{3 + 4i}{3 - 4i} = \frac{(3 + 4i)(3 + 4i)}{(3 - 4i)(3 + 4i)} = \frac{9 + 24i + 16i^2}{9 + 16} = \frac{-7 + 24i}{25}$

Verificamos: $|z| = \left|\frac{-7 + 24i}{25}\right| = \frac{\sqrt{49 + 576}}{25} = \frac{25}{25} = 1$ ✓

$v + \overline{v} = (5 - 12i) + (5 + 12i) = 10$

$\frac{v + \overline{v}}{|v|} = \frac{10}{13}$

$|w|^2 = |7 + 24i|^2 = 7^2 + 24^2 = 49 + 576 = 625$

$w\overline{w} = (7 + 24i)(7 - 24i) = 49 - 576i^2 = 49 + 576 = 625$ ✓

$\frac{u}{v} = \frac{3 + 4i}{5 - 12i} \cdot \frac{5 + 12i}{5 + 12i} = \frac{(3 + 4i)(5 + 12i)}{25 + 144} = \frac{15 + 36i + 20i + 48i^2}{169} = \frac{-33 + 56i}{169}$

$\arg\left(\frac{u}{v}\right) = \arctan\left(\frac{56}{-33}\right) + \pi = \arctan\left(-\frac{56}{33}\right) + \pi$ (segundo cuadrante)

$= \pi - \arctan\left(\frac{56}{33}\right) \approx \pi - 1.038 \approx 2.103$ rad

$|z_1 + z_2|^2 = (a_1 + a_2)^2 + (b_1 + b_2)^2 = a_1^2 + 2a_1a_2 + a_2^2 + b_1^2 + 2b_1b_2 + b_2^2$

$|z_1 - z_2|^2 = (a_1 - a_2)^2 + (b_1 - b_2)^2 = a_1^2 - 2a_1a_2 + a_2^2 + b_1^2 - 2b_1b_2 + b_2^2$

$|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2a_1^2 + 2a_2^2 + 2b_1^2 + 2b_2^2 = 2(|z_1|^2 + |z_2|^2)$ ✓

$z_1\overline{z_2} = (a_1 + b_1i)(a_2 - b_2i) = a_1a_2 + b_1b_2 + i(b_1a_2 - a_1b_2)$

$\text{Re}(z_1\overline{z_2}) = a_1a_2 + b_1b_2 = \text{Re}(z_1)\text{Re}(z_2) + \text{Im}(z_1)\text{Im}(z_2)$ ✓

$|z_1 + z_2|^2 = (z_1 + z_2)\overline{(z_1 + z_2)} = (z_1 + z_2)(\overline{z_1} + \overline{z_2})$

$= z_1\overline{z_1} + z_1\overline{z_2} + z_2\overline{z_1} + z_2\overline{z_2}$

$= |z_1|^2 + |z_2|^2 + z_1\overline{z_2} + \overline{z_1\overline{z_2}}$

$= |z_1|^2 + |z_2|^2 + 2\text{Re}(z_1\overline{z_2})$ ✓

Con $z_1 = 2 + 3i$ y $z_2 = 1 - 4i$:

$|z_1 + z_2|^2 = |3 - i|^2 = 10$, $|z_1 - z_2|^2 = |1 + 7i|^2 = 50$

$2(|z_1|^2 + |z_2|^2) = 2(13 + 17) = 60$

Verificación: $10 + 50 = 60$ ✓

$z_1\overline{z_2} = (2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i^2 = -10 + 11i$

$\text{Re}(z_1\overline{z_2}) = -10 = 2 \cdot 1 + 3 \cdot (-4) = 2 - 12 = -10$ ✓

$|z_1 + z_2|^2 = 13 + 17 + 2(-10) = 10$ ✓