Exercici de números complexos 2 (resolt)

Exercici 1: Donats:

\(z_1 = 3 + 4j\)

\(z_2 = -2 + 5j\)

\(z_3 = 1 - 3j\)

Calcula:

• \(z_1 + z_2\):

  \((3 - 2) + (4 + 5)j = \mathbf{1 + 9j}\) (9.055_83.66º)

• \(z_2 - z_3\):

  \((-2 - 1) + (5 - (-3))j = -3 + (5 + 3)j = \mathbf{-3 + 8j}\) (8.544_110.56º)

Exercici 2: Donats:

\(w_1 = 2 + 3j\)

\(w_2 = 4 - j\)

Calcula:

• \(w_1 \cdot w_2\):

  \((2 \cdot 4) + (2 \cdot -j) + (3j \cdot 4) + (3j \cdot -j) = 8 - 2j + 12j - 3j^2 = 8 + 10j + 3 = \mathbf{11 + 10j}\) (\(14.87_{42.27^\circ}\))

• \(\frac{w_1}{w_2}\):

  \(\frac{2+3j}{4-j} \cdot \frac{4+j}{4+j} = \frac{8+2j+12j+3j^2}{16+1} = \frac{5+14j}{17} = \mathbf{\frac{5}{17} + \frac{14}{17}j} \approx 0,29 + 0,82j\) (\(0,87_{70,54^\circ}\) en polar)

Exercici 3: Donats:

\(z = 5 - 2j\)

\(w = 1 + 2j\)

Calcula

• \(\frac{z}{w}\):

  \(\frac{5-2j}{1+2j} \cdot \frac{1-2j}{1-2j} = \frac{5-10j-2j+4j^2}{1+4} = \frac{5-12j-4}{5} = \frac{1-12j}{5} = \mathbf{\frac{1}{5} - \frac{12}{5}j}\) (2.408_-85.24º)

• \(z \cdot w\):

  \((5 \cdot 1) + (5 \cdot 2j) + (-2j \cdot 1) + (-2j \cdot 2j) = 5 + 10j - 2j - 4j^2 = 5 + 8j + 4 = \mathbf{9 + 8j}\) (12.042_41.63º)

Exercici 4: Donats:

\(z_1 = 4_{30^\circ}\)

\(z_2 = 2_{60^\circ}\)

Calcula:

• \(z_1 \cdot z_2\):

  \((4 \cdot 2)_{30^\circ + 60^\circ} = \mathbf{8_{90^\circ}}\) (\(1,713 + j\))

• \(\frac{z_1}{z_2}\):

  \((\frac{4}{2})_{30^\circ - 60^\circ} = \mathbf{2_{-30^\circ}}\) o \(2_{330^\circ}\) (o \(8j\))

Exercici 5: Donats:

\(z_1 = 2 + 2j\) (Cartesiana)

\(z_2 = 3_{90^\circ}\) (Polar)

Primer, passem \(z_1\) a polar: Mòdul \(r = \sqrt{2^2 + 2^2} = \sqrt{8}\); Angle \(\theta = \arctan(2/2) = 45^\circ\). Per tant, \(z_1 = \sqrt{8}_{45^\circ} \approx 2,83_{45^\circ}\)

En cartesiana: z₂ = 3j

Calcula:

• \(z_1 \cdot z_2\):

  \((\sqrt{8} \cdot 3)_{45^\circ + 90^\circ} = \mathbf{3\sqrt{8}_{135^\circ}} \approx \mathbf{8,48_{135^\circ}}\) (-6 + 6 · j)

• \(\frac{z_1}{z_2}\):

  \((\frac{\sqrt{8}}{3})_{45^\circ - 90^\circ} = \mathbf{(\frac{\sqrt{8}}{3})_{-45^\circ}} \approx \mathbf{0,94_{-45^\circ}}\) (\(\frac{2}{3} - \frac{2}{3}j\) ≈ \(0.66 - 0.66j\))

Exercici 6: Donats:

\(z_1 = 8_{1\,rad}\)

\(z_2 = 9_{0,7\,rad}\)

Calcula:

• \(z_1 \cdot z_2\):

  \((8 \cdot 9)_{1 + 0,7} = \mathbf{72_{1,7\,rad}}\)

• \(\frac{z_1}{z_2}\):

  \((\frac{8}{9})_{1 - 0,7} = \mathbf{(\frac{8}{9})_{0,3\,rad}} \approx \mathbf{0,89_{0,3\,rad}}\)