Complex Cross Product/Examples/3-4i cross -4+3i

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Example of Complex Cross Product

Let:

$z_1 = 3 - 4 i$
$z_2 = -4 + 3 i$

Then:

$z_1 \times z_2 = -7$

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


Proof 1

\(\ds z_1 \times z_2\) \(=\) \(\ds \map \Im {\overline {z_1} z_2}\) Definition 3 of Complex Cross Product
\(\ds \) \(=\) \(\ds \map \Im {\paren {3 + 4 i} \paren {-4 + 3 i} }\) Definition of Complex Conjugate
\(\ds \) \(=\) \(\ds \map \Im {3 \times \paren {-4} - 4 \times 3 + \paren {3 \times 3 + 4 \times \paren {-4} } i}\) Definition of Complex Multiplication
\(\ds \) \(=\) \(\ds 9 + -16\)
\(\ds \) \(=\) \(\ds -7\)


Proof 2

\(\ds z_1 \circ z_2\) \(=\) \(\ds \paren {3 - 4 i} \times \paren {-4 + 3 i}\)
\(\ds \) \(=\) \(\ds 3 \times 3 - \paren {-4} \times \paren {-4}\) Definition 1 of Complex Cross Product
\(\ds \) \(=\) \(\ds 9 - 16\)
\(\ds \) \(=\) \(\ds -7\)

$\blacksquare$