Vector Cross Product is Anticommutative/Complex

Theorem

The complex cross product is anticommutative:

$\forall z_1, z_2 \in \C: z_1 \times z_2 = -\paren {z_2 \times z_1}$

Proof

Let:

$z_1 := x_1 + i y_1, z_2 = x_2 + i y_2$

Then:

\(\ds z_1 \times z_2\)

\(=\)

\(\ds x_1 y_2 - y_1 x_2\)

Definition 1 of Complex Cross Product

\(\ds \)

\(=\)

\(\ds -\paren {x_2 y_1 - y_2 x_1}\)

Real Addition is Commutative and Real Multiplication is Commutative

\(\ds \)

\(=\)

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

Definition 1 of Complex Cross Product

$\blacksquare$

Examples

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

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

Let:

$z_1 = 2 + 5 i$

$z_2 = 3 - i$

Then:

$z_1 \times z_2 = -17$

where $\times$ denotes (complex) cross product.

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

Let:

$z_1 = 3 - i$

$z_2 = 2 + 5 i$

Then:

$z_1 \times z_2 = 17$

where $\times$ denotes (complex) cross product.

As can be seen:

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

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The Dot and Cross Product: $111 \ \text{(b)}$