# 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

$\blacksquare$