The three animations demonstrate that the vector triple product $a \times ( b \times c )$ is not merely a sequence of rotations, but a structured projection into a specific plane. The first animation establishes geometric consistency, showing that the result is always trapped within the plane defined by $b$ and $c$ because it must be orthogonal to their normal. The second animation highlights computational efficiency, revealing how the "bac-cab" rule simplifies work by showing that terms "vanish" when $a$ is orthogonal to either $b$ or $c$, reducing a complex nested cross-product into simple scaling. Finally, the third animation proves the universal scaling of the identity using non-unit vectors; it visually decomposes the result into a vector sum of two "ghost" components, proving that the $\varepsilon-\delta$ relation holds true regardless of the vectors' magnitudes or orientations.
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%% Condensed Notes
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%% Proof and Derivation
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%% %% Condensed Notes
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%% Proof and Derivation
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