Category:Examples of Complex Cross Product
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This category contains examples of Complex Cross Product.
Let $z_1 := x_1 + i y_1$ and $z_2 := x_2 + i y_2$ be complex numbers.
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.
Pages in category "Examples of Complex Cross Product"
The following 8 pages are in this category, out of 8 total.
C
- Complex Cross Product/Examples
- Complex Cross Product/Examples/2+5i cross 3-i
- Complex Cross Product/Examples/3-4i cross -4+3i
- Complex Cross Product/Examples/3-4i cross -4+3i/Proof 1
- Complex Cross Product/Examples/3-4i cross -4+3i/Proof 2
- Complex Cross Product/Examples/3-i cross 2+5i
- Complex Cross Product/Examples/Size of 2+5i cross 3-i
- Complex Cross Product/Examples/Size of 3-i cross 2+5i