Question

Dados os números complexos Z1=1+3i e Z2= -2+i , Calcular :

a) Z1.Z2 b) Z1/Z2

Answer 1

a)
$z_{1} \cdot z_{2}=\left(1+3i \right )\cdot\left(-2+i \right )\\ \\ =1\cdot\left(-2 \right )+1\cdot i+3i\cdot\left(-2 \right )+3i\cdot i\\ \\ =-2+i-6i+3i^{2}\\ \\ =-2-5i+3\cdot\left(-1 \right )\\ \\ =-2-3-5i\\ \\ =\boxed{-5-5i}$


b)
$\frac{z_{1}}{z_{2}}=\frac{1+3i}{-2+i}\\ \\ =\frac{\left(1+3i \right )\cdot \left(-2-i \right )}{\left(-2+i \right )\left(-2-i \right )}\\ \\ =\frac{1\cdot\left(-2 \right )+1\cdot\left(-i \right )+3i\cdot\left(-2 \right )+3i\cdot \left(-i \right )}{\left(-2 \right )\cdot\left(-2 \right )-2\cdot(-i)+i\cdot\left(-2 \right )+i\cdot(-i)}\\ \\ =\frac{-2-i-6i-3i^{2}}{4+2i-2i-i^{2}}\\ \\ =\frac{-2-7i-3\cdot(-1)}{4-(-1)}\\ \\ =\frac{-2+3-7i}{4+1}\\ \\ =\frac{1-7i}{5}\\ \\ =\boxed{\frac{1}{5}-\frac{7}{5}i}$