BAC-CAB Algebraic and Geometric Proofs (BCA)

The BAC-CAB rule serves as a vital identity in vector calculus, expressing the vector triple product $\vec{a} \times(\vec{b} \times \vec{c})$ as the specific linear combination $\vec{b}(\vec{a} \cdot \vec{c}) - \vec{c}(\vec{a} \cdot \vec{b})$. According to the sources, this identity is mathematically grounded in the $\varepsilon-\delta$ relation, which relates the product of two Levi-Civita symbols to a combination of Kronecker deltas. To derive the rule efficiently, one should utilise the symmetries and anti-symmetries of these expressions rather than performing explicit sums for every possible index. Furthermore, the identity can be visually verified through a systematic five-step construction that tracks individual vectors using distinct colours and line styles to demonstrate that the left-hand and right-hand sides ultimately result in the same vector. While mnemonics are a common tool to remember this expansion, our conversation highlights that one should remain attentive to the correct sequence of vectors to ensure the mathematical accuracy of the final expression.

Analogy to solidify understanding: The $\varepsilon-\delta$ relation is like the gears inside a watch; you don't necessarily need to count every tooth on every gear to understand how they turn, provided you understand the symmetry of how they interlock. Once those gears are in place, the hands of the watch (the BAC-CAB rule) will always point to the correct result.

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🏗️Structural clarification of Poof and Derivation

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