Complex Multiplication: Rotation & Scaling

Overview

A short animation that visualizes multiplication of two complex numbers on the complex plane, illustrating how the operation combines a rotation (by the argument of the multiplier) and a scaling (by its modulus). Viewers see a vector representing a base complex number being transformed step‑by‑step as it is multiplied by a second complex number.

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Phases

#	Phase Name	Duration	Description
1	Intro	~2 s	Title fades in, then the complex plane (axes) appears with a grid. A red vector $z = 1 + i$ is drawn from the origin.
2	Show Multiplier	~3 s	A blue vector $w = \sqrt{2}\,e^{i\pi/4}$ (i.e., $w = 1 + i$) is introduced, its modulus and argument are highlighted with a radial arc and a small arc‑length label.
3	Scaling & Rotation	~5 s	The red vector $z$ animates: first it scales by $|w|$ (length doubles), then it rotates by $\arg w$ (45°) around the origin, ending at the product $zw$. The path is traced with a faint line.
4	Product Reveal	~3 s	The resulting vector $zw$ is highlighted in green, and the algebraic result $zw = (1+i)(1+i) = 2i$ appears briefly in the bottom area.
5	General Formula	~4 s	A generic illustration: a generic vector $z$ and a generic multiplier $w = re^{i\theta}$ are shown, with arrows indicating scaling by $r$ and rotation by $\theta$. The formula $zw = |z|\,|w|\,e^{i(\arg z+\arg w)}$ fades in.
6	Outro	~2 s	The scene holds the final frame (all elements remain on screen) instead of fading out.

Total runtime: ~19 seconds (under the 20‑second target).

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Layout

┌─────────────────────────────────────────────┐
│                 TOP AREA                    │
├──────────────────────┬──────────────────────┤
│                      │                      │
│     LEFT AREA        │     RIGHT AREA       │
│  (Complex plane)     │ (Optional labels)   │
│                      │                      │
├──────────────────────┴──────────────────────┤
│                 BOTTOM AREA                 │
└─────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Complex Multiplication: Rotation & Scaling" (fades in)	Appears in Intro and remains visible through the end
Left	Main visual: complex plane with axes, grid, vectors $z$, $w$, $zw$	Primary focus throughout
Right	Small auxiliary graphics: radial arc for $\arg w$, modulus bar for $|w|$	Appears in Phase 2 and 5
Bottom	Brief algebraic expressions (e.g., $zw = 2i$, generic formula)	Small font, fades in with relevant phase

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Notes

• Runtime: Keep total runtime under 20 seconds (current estimate ~19 s).
• Use smooth linear interpolation for scaling, then a smooth rotation (or combined scaling‑rotation via a single homothety‑rotation animation).
• Traced path of the moving vector should be semi‑transparent to emphasize the transformation.
• No textual narration; all information conveyed visually via vectors, arcs, and brief equations.
• The scene must be a single Manim Scene class.
• The final frame is held (no fade‑out) as per the updated request.