First slide

Geometry of complex numbers

Question

Area of a triangle with vertices given by $z, i z$ $z + i z,$ where $z$ is a complex number, is

Moderate

A

0
B

$\frac{1}{2} | z |^{2}$
C

$| z |^{2}$
D

$2 | z |^{2}$

Solution

Area of triangle is
$Δ = \frac{1}{4} | | \begin{array}{ccc} z & \bar{z} & 1 \\ i z & - i \bar{z} & 1 \\ z + i z & \bar{z} - i \bar{z} & 1 \end{array} ∣$
Using $R_{3} \to R_{3} - R_{1} - R_{2}$ we get
$Δ = \frac{1}{4} | | \begin{array}{rrr} z \bar{z} 1 \\ i z - i \bar{z} 1 \\ 0 0 - 1 \end{array} ∣$
$= \frac{1}{4} | - i z \bar{z} - i z \bar{z} | = \frac{1}{2} | z |^{2}$

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