Complex Dot Product/Examples/3-4i dot -4+3i/Proof 2

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

\(\ds z_1 \circ z_2\)

\(=\)

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

\(\ds \)

\(=\)

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

Definition 1 of Dot Product

\(\ds \)

\(=\)

\(\ds -12 + -12\)

\(\ds \)

\(=\)

\(\ds -24\)

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Dot and Cross Product: $39 \ \text{(a)}$