Complex Arithmetic/Examples/Modulus of (3 z 1 - 4 z 2)

Example of Complex Arithmetic

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

Then:

$\cmod {3 z_1 - 4 z_2} = \sqrt {157}$

Proof

\(\ds 3 z_1 - 4 z_2\)

\(=\)

\(\ds 3 \paren {2 + i} - 4 \paren {3 - 2 i}\)

\(\ds \)

\(=\)

\(\ds \paren {6 + 3 i} - \paren {12 - 8 i}\)

\(\ds \)

\(=\)

\(\ds -6 + 11 i\)

\(\ds \leadsto \ \ \)

\(\ds \cmod {3 z_1 - 4 z_2}\)

\(=\)

\(\ds \sqrt {\paren {-6}^2 + \paren {11}^2}\)

Definition of Complex Modulus

\(\ds \)

\(=\)

\(\ds \sqrt {36 + 121}\)

\(\ds \)

\(=\)

\(\ds \sqrt {157}\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Fundamental Operations with Complex Numbers: $2 \ \text{(a)}$