Complex Dot Product/Examples/3-4i dot -4+3i/Proof 1
Jump to navigation
Jump to search
Examples of Complex Dot Product
Let:
- $z_1 = 3 - 4 i$
- $z_2 = -4 + 3 i$
Then:
- $z_1 \circ z_2 = -24$
where $\circ$ denotes (complex) dot product.
Proof
\(\ds z_1 \circ z_2\) | \(=\) | \(\ds \map \Re {\overline {z_1} z_2}\) | Definition 3 of Dot Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {\paren {3 + 4 i} \paren {-4 + 3 i} }\) | Definition of Complex Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Re {3 \times \paren {-4} - 4 \times 3 + \paren {3 \times 3 + 4 \times \paren {-4} } i}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds -12 + -12\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -24\) |
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Dot and Cross Product: $39 \ \text{(a)}$