Complex Dot Product/Examples/3-4i dot -4+3i

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 1

\(\ds z_1 \circ z_2\)

\(=\)

\(\ds \map \Re {\overline {z_1} z_2}\)

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

\(\ds \)

\(=\)

\(\ds -12 + -12\)

\(\ds \)

\(=\)

\(\ds -24\)

$\blacksquare$

Proof 2

\(\ds z_1 \circ z_2\)

\(=\)

\(\ds \paren {3 - 4 i} \circ \paren {-4 + 3 i}\)

\(\ds \)

\(=\)

\(\ds 3 \times \paren {-4} + \paren {-4} \times 3\)

Definition 1 of Dot Product

\(\ds \)

\(=\)

\(\ds -12 + -12\)

\(\ds \)

\(=\)

\(\ds -24\)

$\blacksquare$