# Complex Cross Product Distributes over Addition

## Theorem

Let $z_1, z_2, z_3 \in \C$ be complex numbers.

Then:

$z_1 \times \paren {z_2 + z_3} = z_1 \times z_2 + z_1 \times z_3$

where $\times$ denotes cross product.

## Proof

Let:

$z_1 = x_1 + i y_1$
$z_2 = x_2 + i y_2$
$z_3 = x_3 + i y_3$

Then:

 $\displaystyle z_1 \times \paren {z_2 + z_3}$ $=$ $\displaystyle \paren {x_1 + i y_1} \times \paren {\paren {x_2 + i y_2} + \paren {x_3 + i y_3} }$ $\displaystyle$ $=$ $\displaystyle \paren {x_1 + i y_1} \times \paren {\paren {x_2 + x_3} + i \paren {y_2 + y_3} }$ Definition of Complex Addition $\displaystyle$ $=$ $\displaystyle x_1 \left({y_2 + y_3}\right) - y_1 \left({x_2 + x_3}\right)$ Definition 1 of Complex Cross Product $\displaystyle$ $=$ $\displaystyle x_1 y_2 + x_1 y_3 - y_1 x_2 - y_1 x_3$ Real Multiplication Distributes over Addition $\displaystyle$ $=$ $\displaystyle x_1 y_2 - y_1 x_2 + x_1 y_3 - y_1 x_3$ Real Addition is Commutative $\displaystyle$ $=$ $\displaystyle z_1 \times z_2 + z_1 \times z_3$ Definition 1 of Complex Cross Product

$\blacksquare$