Complex Modulus/Examples/2 conj z1 - 3 conj z2 - 2

From ProofWiki
Jump to navigation Jump to search

Example of Complex Modulus

Let $z_1 = 4 - 3 i$ and $z_2 = -1 + 2 i$.

Then:

$\cmod {2 \overline {z_1} - 3 \overline {z_2} - 2} = 15$


Proof 1

An illustration of the evaluation of $\cmod {2 \overline {z_1} - 3 \overline {z_2} - 2}$, where:

$z_1 = 4 - 3 i$
$z_2 = -1 + 2 i$

is given below:


Complex-Conjugate-2 conj z1 - 3 conj z2 - 2.png


The modulus is seen to be the radius of the circle.

$\blacksquare$


Proof 2

\(\ds \cmod {2 \overline {z_1} - 3 \overline {z_2} - 2}\) \(=\) \(\ds \cmod {2 \paren {\overline {4 - 3 i} } - 3 \paren {\overline {-1 + 2 i} } - 2}\) Definition of $z_1$ and $z_2$
\(\ds \) \(=\) \(\ds \cmod {2 \paren {4 + 3 i} - 3 \paren {-1 - 2 i} - 2}\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds \cmod {\paren {8 + 6 i} - \paren {-3 - 6 i} - 2}\)
\(\ds \) \(=\) \(\ds \cmod {\paren {8 - \paren {-3} } + \paren {6 - \paren {-6} } i - 2}\) Definition of Complex Subtraction
\(\ds \) \(=\) \(\ds \cmod {11 + 12 i - 2}\)
\(\ds \) \(=\) \(\ds \cmod {9 + 12 i}\)
\(\ds \) \(=\) \(\ds \sqrt {9^2 + 12^2}\)
\(\ds \) \(=\) \(\ds 3 \sqrt {3^2 + 4^2}\)
\(\ds \) \(=\) \(\ds 3 \sqrt {9 + 15}\)
\(\ds \) \(=\) \(\ds 3 \sqrt {25}\)
\(\ds \) \(=\) \(\ds 3 \times 5\)
\(\ds \) \(=\) \(\ds 15\)

$\blacksquare$