First slide

Algebra of complex numbers

Question

The area of the triangle whose vertices are the points represented by the complex number $z, i z$ and $z i + z$ is

Moderate

A

$\frac{1}{4} | z |^{2}$
B

$\frac{1}{8} | z |^{2}$
C

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

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

Solution

Area of the triangle is given by
$Δ = \frac{1}{4} |\begin{array}{ccc} z & \bar{z} & 1 \\ i z & - i \bar{z} & 1 \\ z + i z & \bar{z} - i \bar{z} & 1 \end{array}|$
Applying $R_{3} \to R_{3} - R_{1} - R_{2}$ we get
$\begin{aligned} Δ = \frac{1}{4} |\begin{array}{ccc} z & \bar{z} & 1 \\ i z & - i \bar{z} & 1 \\ 0 & 0 & - 1 \end{array}| |= \frac{1}{4}| (- 1) |\begin{array}{cc} z & \bar{z} \\ i z & - i \bar{z} \end{array}| ∣ \\ = \frac{1}{4} | (- 1) (i) z \bar{z} | |\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}| = \frac{1}{4} | z |^{2} (2) = \frac{1}{2} | z |^{2} \end{aligned}$

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