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:
\(\ds z_1 \times \paren {z_2 + z_3}\) | \(=\) | \(\ds \paren {x_1 + i y_1} \times \paren {\paren {x_2 + i y_2} + \paren {x_3 + i y_3} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x_1 + i y_1} \times \paren {\paren {x_2 + x_3} + i \paren {y_2 + y_3} }\) | Definition of Complex Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds x_1 \left({y_2 + y_3}\right) - y_1 \left({x_2 + x_3}\right)\) | Definition 1 of Complex Cross Product | |||||||||||
\(\ds \) | \(=\) | \(\ds x_1 y_2 + x_1 y_3 - y_1 x_2 - y_1 x_3\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds x_1 y_2 - y_1 x_2 + x_1 y_3 - y_1 x_3\) | Real Addition is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds z_1 \times z_2 + z_1 \times z_3\) | Definition 1 of Complex Cross Product |
$\blacksquare$
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: $113 \ \text{(b)}$