Chapter A.7, Problem 90AYU

To determine

To show: The equations are $\overline{z\times w}=\overline{z}\times \overline{w}$ .

Explanation of Solution

Given information:

The complex numberis $z=a+ib,w=c+di$ .

Calculation:

Consider the equation.

  $\begin{array}{c}\overline{z\times w}=\overline{\left(a+bi\right)\left(c-di\right)}\\ =\overline{ac+\left(ad\right)i+\left(bc\right)i+bd{i}^2}\\ =\overline{ac+adi+bci-bd}\\ =\overline{\left(ac-bd\right)-\left(ad+bc\right)}i\\ =\left(ac-bd\right)-\left(ad+bc\right)i\end{array}$

Consider the equation.

  $\begin{array}{c}\overline{z}\times \overline{w}=\overline{\left(a+bi\right)}\times \overline{\left(c-di\right)}\\ =\left(a-bi\right)\left(c+di\right)\\ =ac+a\left(-d\right)i-bci+bd{i}^2\\ =ac-adi-bci-bd\\ =\left(ac-bd\right)-\left(ad+bc\right)i\end{array}$