Question

Sendo Z1 = (1/2+i) e Z2 = (1/2-i). Determine:

a) O conjugado de Z1 . Z2
b) Z1 : Z2

Answer 1

a)
$Z_{1} .Z_{2} =(\frac{1}{2}+i).( \frac{1}{2} -i) \\ \\ Z_{1} .Z_{2} =( \frac{1}{2})^{2} - i^{2} \\\\Z_{1} .Z_{2} = \frac{1}{4} -(-1)\\\\Z_{1} .Z_{2}= \frac{1}{4} +1\\\\Z_{1} .Z_{2} = \boxed{\frac{5}{4}}$

b)
$\frac{Z_{1} }{Z_{2}} = \frac{( \frac{1}{2}+i) }{( \frac{1}{2}-i)} . \frac{( \frac{1}{2}+i)}{( \frac{1}{2}+i)} = \frac{( \frac{1}{4}+ \frac{1}{2}i + \frac{1}{2}i+i^{2} )}{ \frac{5}{4}} = \frac{ -\frac{3}{4}i }{ \frac{5}{4} }=\boxed{- \frac{3}{5}i}$