Complex Modulus/Examples/2 conj z1 - 3 conj z2 - 2
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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:
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {11 + 12 i - 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {9 + 12 i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {9^2 + 12^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sqrt {3^2 + 4^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sqrt {9 + 15}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sqrt {25}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \times 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 15\) |
$\blacksquare$