📢BAC-CAB Algebraic and Geometric Proofs

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.

🎬Narrated Video

Description

This illustration, titled "A Visual Guide to the BAC-CAB Rule," provides a comprehensive overview of the vector triple product and the mathematical tools used to prove it.

The BAC-CAB Identity

The central focus is the formula for the vector triple product:

$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b}(\vec{a} \cdot \vec{c}) - \vec{c}(\vec{a} \cdot \vec{b})$

Transformation: The rule is described as transforming a complex triple cross product into a simpler linear combination of vectors.
Mnemonic: It features the "BAC - CAB" mnemonic, showing how the letters correspond to the vectors $\vec{b}, \vec{a}, \vec{c}$ and $\vec{c}, \vec{a}, \vec{b}$ in the resulting expression to help with memorization.

Visual and Formal Proof Tools

The right side of the illustration breaks down how the identity is understood geometrically and derived formally:

Visual Proof

It provides a geometric representation using 3D coordinate axes.
The visualization demonstrates that the resulting vector is equal to the linear combination of $\vec{b}$ and $\vec{c}$ .

The Formal Proof Tool

Levi-Civita ( $\epsilon-\delta$ ) Relation: This section introduces the identity used for formal derivation:
$\epsilon_{ijk}\epsilon_{klm} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}$
Purpose: This tensor identity is the foundational tool used to derive the BAC-CAB rule from its individual vector components.