Vector Addition and Subtraction

Overview

An animation that visually demonstrates how to add and subtract two vectors in the plane using the tip‑to‑tail method and component-wise arithmetic. Viewers see the geometric construction, the resulting vector, and the corresponding algebraic expressions, reinforcing the equivalence of graphical and component approaches.

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Phases

#	Phase Name	Duration	Description
1	Intro	~3s	Title "Vector Addition & Subtraction" fades in at the top. Two vectors $\mathbf{a}$ and $\mathbf{b}$ appear in the left area, each labeled with their components (e.g., $\mathbf{a}= (2,1)$, $\mathbf{b}= (1,3)$).
2	Addition – Tip‑to‑Tail	~7s	Vector $\mathbf{a}$ is drawn from the origin. Then $\mathbf{b}$ is placed tip‑to‑tail of $\mathbf{a}$. The resultant vector $\mathbf{a}+\mathbf{b}$ is drawn from the origin to the tip of $\mathbf{b}$. Component formula $(2,1)+(1,3) = (3,4)$ appears in the bottom area.
3	Addition – Component View	~5s	The same vectors are shown originating from the origin side‑by‑side. Dashed horizontal and vertical lines illustrate component addition, culminating in the same resultant vector highlighted in the same color as before.
4	Subtraction – Tip‑to‑Tail	~7s	Vector $\mathbf{b}$ is drawn from the origin. Then $-\mathbf{a}$ (the reverse of $\mathbf{a}$) is placed tip‑to‑tail of $\mathbf{b}$. The resultant vector $\mathbf{b}-\mathbf{a}$ is drawn from the origin. Component formula $(1,3)-(2,1) = (-1,2)$ appears in the bottom area.
5	Subtraction – Component View	~5s	Parallel component illustration similar to addition, showing subtraction of each component and the resulting vector.
6	Outro	~3s	All vectors fade out, leaving a concise summary: "Tip‑to‑tail and component methods give the same result." The title fades out.

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Layout

┌───────────────────────────────────────┐
│               TOP AREA                 │
├───────────────────────┬───────────────┤
│        LEFT AREA       │   RIGHT AREA │
│  (main vector graphics │  (optional    │
│   and constructions)  │   brief notes)│
├───────────────────────┴───────────────┤
│               BOTTOM AREA              │
└───────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Title "Vector Addition & Subtraction"	Fades in at start of Intro, stays until Outro
Left	Primary visual: vectors, tip‑to‑tail constructions, component grids	Central focus; colors differentiate $\mathbf{a}$, $\mathbf{b}$, and results
Right	Minimal supporting text: brief labels like "Resultant" or "Negative $\mathbf{a}$" when needed	Appears only in phases that require clarification
Bottom	Algebraic component equations for each operation	Fades in with the corresponding phase, fades out before next phase

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Notes

• Use a consistent color scheme: e.g., $\mathbf{a}$ – blue, $\mathbf{b}$ – orange, resultant – green, negative $\mathbf{a}$ – light blue.
• All motions should be smooth but quick to keep total runtime under 30 seconds.
• No textual narration is required; visual cues and brief on‑screen equations convey the concepts.
• The scene must be implemented as a single Manim Scene class.
• Avoid any additional text that does not directly support the visual explanation.