Equivalence of Definitions of Complex Cross Product
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Theorem
The following definitions of the concept of Complex Cross Product are equivalent:
Definition 1
The cross product of $z_1$ and $z_2$ is defined as:
- $z_1 \times z_2 = x_1 y_2 - y_1 x_2$
Definition 2
The cross product of $z_1$ and $z_2$ is defined as:
- $z_1 \times z_2 = \cmod {z_1} \, \cmod {z_2} \sin \theta$
where:
- $\cmod {z_1}$ denotes the complex modulus of $z_1$
- $\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.
Definition 3
The cross product of $z_1$ and $z_2$ is defined as:
- $z_1 \times z_2 := \map \Im {\overline {z_1} z_2}$
where:
- $\map \Im z$ denotes the imaginary part of a complex number $z$
- $\overline {z_1}$ denotes the complex conjugate of $z_1$
- $\overline {z_1} z_2$ denotes complex multiplication.
Definition 4
The cross product of $z_1$ and $z_2$ is defined as:
- $z_1 \times z_2 := \dfrac {\overline {z_1} z_2 - z_1 \overline {z_2}} {2 i}$
where:
- $\overline {z_1}$ denotes the complex conjugate of $z_1$
- $\overline {z_1} z_2$ denotes complex multiplication.
Proof
Definition 1 equivalent to Definition 3
\(\ds \) | \(\) | \(\ds \map \Im {\overline {z_1} z_2}\) | Definition 3 of Vector Cross Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\paren {x_1 - i y_1} \paren {x_2 + i y_2} }\) | Definition of Complex Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Im {\paren {x_1 x_2 + y_1 y_2} + i \paren {x_1 y_2 - x_2 y_1} }\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds x_1 y_2 - x_2 y_1\) | Definition of Imaginary Part |
$\Box$
Definition 2 equivalent to Definition 3
\(\ds \) | \(\) | \(\ds \map \Im {\overline {z_1} z_2}\) | Definition 3 of Vector Cross Product | |||||||||||
\(\ds \) | \(=\) | \(\ds r_1 r_2 \sin \paren {\theta_2 - \theta_1}\) | Complex Cross Product in Exponential Form | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {z_1} \, \cmod {z_2} \map \sin {\theta_2 - \theta_1}\) | Definition of Polar Form of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {z_1} \, \cmod {z_2} \sin \theta\) | where $\theta = \theta_2 - \theta_1$ is the angle between $z_1$ and $z_2$ |
$\Box$
Definition 1 equivalent to Definition 4
\(\ds \) | \(\) | \(\ds \frac {\overline {z_1} z_2 - z_1 \overline {z_2} } {2 i}\) | Definition 4 of Complex Dot Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {x_1 - i y_1} \paren {x_2 + i y_2} - \paren {x_1 + i y_1} \paren {x_2 - i y_2} } {2 i}\) | Definition of Complex Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\paren {x_1 x_2 + y_1 y_2} + i \paren {x_1 y_2 - x_2 y_1} } - \paren {\paren {x_1 x_2 + y_1 y_2} + i \paren {-x_1 y_2 + x_2 y_1} } } {2 i}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds x_1 y_2 - x_2 y_1\) | after algebra |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Dot and Cross Product