Complex Dot Product/Examples/3-4i dot -4+3i/Acute Angle Between

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Examples of Complex Dot Product

Consider:

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

expressed as vectors.

Then the acute angle between $z_1$ and $z_2$ is approximately $16 \degrees 16'$


Proof

Let $\theta$ denote the acute angle between $z_1$ and $z_2$.

Then:

\(\ds z_1 \circ z_2\) \(=\) \(\ds \cmod {z_1} \cmod {z_2} \cos \theta\) Definition 2 of Dot Product
\(\ds \leadsto \ \ \) \(\ds \cos \theta\) \(=\) \(\ds \dfrac {z_1 \circ z_2} {\cmod {z_1} \cmod {z_2} }\)
\(\ds \) \(=\) \(\ds \dfrac {-24} {\cmod {z_1} \cmod {z_2} }\) Complex Dot Product of $3 - 4 i$ and $-4 + 3 i$
\(\ds \) \(=\) \(\ds \dfrac {-24} {\sqrt {3^2 + \paren {-4}^2} \sqrt {3^2 + \paren {-4}^2} }\) Definition of Complex Modulus
\(\ds \) \(=\) \(\ds \dfrac {-24} {25}\)
\(\ds \) \(=\) \(\ds -0 \cdotp 96\)
\(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds 16 \degrees 16'\) by calculation, or looking up tables

$\blacksquare$


Sources