Determine and use complex conjugates.

Let \(z_1 = 3 + 2i\) and \(z_2 = 5 - 4i\). (a) Determine \(\overline{z_1}\) and \(\overline{z_2}\). (b) Use your results from part (a) to determine \(\overline{z_1} \times \overline{z_2}\). (c) Hence determine \(\overline{z_1 \times z_2}\) and verify that \(\overline{z_1 \times z_2} = \overline{z_1} \times \overline{z_2}\).

The complex number \(z\) and its conjugate \(\overline{z}\) are shown on the Argand diagram. Determine \(z - \overline{z}\).

Let \(w = 3 - 2i\) and \(z = 4 + 7i\). (a) Determine \(\bar{w}\) and \(\bar{z}\). (1 mark) (b) Determine \(\bar{w} + \bar{z}\). (1 mark) (c) Use your results from parts (a) and (b) to verify that \(\overline{w + z} = \bar{w} + \bar{z}\). (1 mark)

An electrical engineer is analysing the impedance in an alternating current circuit. The total impedance \(Z\) (measured in ohms) is given by \(Z = \frac{w}{w - \bar{w}}\), where \(w = 3 + 7i\) represents a component's complex impedance and \(\bar{w}\) is the complex conjugate of \(w\). (a) Determine the complex conjugate \(\bar{w}\). (1 mark) (b) Use your result from part (a) to determine the total impedance \(Z\) in the form \(a + bi\). (2 marks)