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 | |||||||||||
\(\ds \) | \(=\) | \(\ds 9 - 16\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -7\) |
$\blacksquare$