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
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Dot and Cross Product: $40$