# 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. ### :clapper:Narrated Video {% embed url="" %} ### :thread:Related Derivation {% content-ref url="../proof-and-derivation/proving-the-epsilon-delta-relation-and-the-bac-cab-rule" %} [proving-the-epsilon-delta-relation-and-the-bac-cab-rule](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/proof-and-derivation/proving-the-epsilon-delta-relation-and-the-bac-cab-rule) {% endcontent-ref %} ### :hammer\_pick:Compound Page {% embed url="" %}