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Modulus of a Complex Number

If z1, z2, ...

Question

If $z_{1}, z_{2}, z_{3}$ are three complex numbers such that $5 z_{1} - 13 z_{2} + 8 z_{3} = 0$ , then show that
$∣ ∣ ∣ ∣ ∣ \begin{matrix} z_{1} & {¯ ¯ ¯ z}_{1} & 1 z_{2} & {¯ ¯ ¯ z}_{2} & 1 z_{3} & {¯ ¯ ¯ z}_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$

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Solution

$5 z_{1} - 13 z_{2} + 8 z_{3} = 0$
or $\frac{5 z_{1} + 8 z_{2}}{5 + 8} = z_{3}$
Hence, $z_{1}, z_{2}, z_{3}$ are collinear.
$∣ ∣ ∣ ∣ ∣ \begin{matrix} z_{1} & {¯ ¯ ¯ z}_{1} & 1 z_{2} & {¯ ¯ ¯ z}_{2} & 1 z_{3} & {¯ ¯ ¯ z}_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ =$ 0 (condition of collinear points)

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are three complex numbers such that

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∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = 0

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