Absolute Value of Complex Cross Product is Commutative

Theorem

Let $z_1$ and $z_2$ be complex numbers.

Let $z_1 \times z_2$ denote the (complex) cross product of $z_1$ and $z_2$.

Then:

$\size {z_1 \times z_2} = \size {z_2 \times z_1}$

where $\size {\, \cdot \,}$ denotes the absolute value function.

Proof

\(\ds \size {z_2 \times z_1}\)

\(=\)

\(\ds \size {-z_1 \times z_2}\)

Complex Cross Product is Anticommutative

\(\ds \)

\(=\)

\(\ds \size {z_1 \times z_2}\)

Definition of Absolute Value

Hence the result.

$\blacksquare$

Examples

Example: $\size {\paren {2 + 5 i} \times \paren {3 - i} } = \size {\paren {3 - i} \times \paren {2 + 5 i} }$

Example: $\size {\paren {2 + 5 i} \times \paren {3 - i} }$

Let:

$z_1 = 2 + 5 i$

$z_2 = 3 - i$

Then:

$\size {z_1 \times z_2} = 17$

Example: $\size {\paren {3 - i} \times \paren {2 + 5 i} }$

Let:

$z_1 = 3 - i$

$z_2 = 2 + 5 i$

Then:

$\size {z_1 \times z_2} = 17$

As can be seen:

$\size {\paren {2 + 5 i} \times \paren {3 - i} } = \size {\paren {3 - i} \times \paren {2 + 5 i} }$

$\blacksquare$