Complex Cross Product Distributes over Addition

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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

$\blacksquare$


Sources