Equation of Line in Complex Plane/Formulation 2
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Theorem
Let $\C$ be the complex plane.
Let $L$ be the infinite straight line in $\C$ which is the locus of the equation:
- $l x + m y = 1$
Then $L$ may be written as:
- $\map \Re {a z} = 1$
where $a$ is the point in $\C$ defined as:
- $a = l - i m$
Proof
Let $z = x + i y$.
Let $a = l - i m$.
Then:
\(\ds \map \Re {a z}\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\paren {a z + \overline {a z} } } 2\) | \(=\) | \(\ds 1\) | Sum of Complex Number with Conjugate | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a z + \overline a \cdot \overline z\) | \(=\) | \(\ds 2\) | Complex Modulus of Product of Complex Numbers | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {l - i m} \paren {x + i y} + \paren {l + i m} \paren {x - i y}\) | \(=\) | \(\ds 2\) | Definition of Complex Conjugate | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\paren {l x + m y} + i \paren {l y - m x} } + \paren {\paren {l x + m y} - i \paren {l y - m x} }\) | \(=\) | \(\ds 2\) | Definition of Complex Multiplication | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds l x + m y\) | \(=\) | \(\ds 1\) | simplifying |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Example $4$: $(2.6)$