Magnitude of Projection of Complex Number on Another
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Theorem
Let $z_1$ and $z_2$ denote complex numbers in vector form.
Let $\map {\pr_1} {z_1, z_2}$ denote the projection of $z_1$ on $z_2$.
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Then:
- $\cmod {\map {\pr_1} {z_1, z_2} } = \dfrac {\cmod {z_1 \circ z_2} } {\cmod {z_2} }$
where:
- $z_1 \circ z_2$ denotes complex dot product
- $\cmod {z_2}$ denotes complex modulus.
Proof
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Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Dot and Cross Product: $3.$